I’m currently running a survival sandbox game with 4 players(Lizardfolk Ranger, Human Ranger, Simic Hybrid Sorcerer, Dwarf Barbarian) and I’m having trouble coming up with rulings on how travel and structure building would work. The setting involves an apocalyptic event that made all of the monsters go insane, and wiped out most of civilization. I mostly use random encounters when they travel for more then 4 miles at a time, but that is getting extremely boring and mostly just turns the session into a slow, irritating slog.

Additionally, my sorcerer tried building a structure as a base, but I didn’t know how to rule using naturally foraged materials to build a structure, and I want it to be more then just a skill check.

Any ideas?

Probably play a TTRPG that isn't DND, as a start lol. But joking aside, I'd look to DMSGuild or other third party products to help you with this. 

Matt Colville's Strongholds and Followers book could give you the inspiration you're looking for to do "structure building". 

But seriously, I'm gonna reiterate this again - I've played in a survival style DND game that wasn't explicitly Western Marches and it was incredibly frustrating. Every utility that SHOULD make the survival aspect easy (Create Water, backgrounds that elt you forage for food, proficiency in sailing, etc) seemed to get removed outright or nerfed until they're unusable. The game left a bad taste in my mouth, personally. I'd just say to make sure that your players WANT to play in a game like you're describing. If so, then have at it and good luck. 

Thanks for the advice!

For the survival part, i would recommend having a solid plan for the area the characters are in. This doesn't have to be the whole kingdom, just the immediate area from the players. This allows you to know what creatures will be in the area, and how you can create encounters. E.g you know that ettercaps and spiders live in the forest, meaning if the players ever go through it then you can create an encounter with them, and make it meaningful to the story.

Remember that difficult terrain exists, and it could be that the players run into a patch of swampland that is ruled as difficult terrain: this means that they have to choose between going through it: but being slowed dramatically, or finding/planning a different route, or even going to somewhere else (if its a fetch quest and the item can be found in multiple places). It also opens up the possibility for unique spells or abilities, and helps the rangers feel special by being able to say "its my favoured terrain, we can pass through fine!". Thats another point: read through any class/race abilities that could be applicable to travel and make scenes for these abilities to come into play (rangers especially).

As far as building is concerned, i believe that xanathars has some rules on it. If not, then there are plenty of free supplements on the internet as well as videos on youtube. They generally tell you that they need to aquire materials (which includes going to towns or villages and bartering with traders), for which the material cost may not be just gold, or the characters may not have enough gold, leading to more quests in order to recieve/find some. Using ability checks is useful, but dont overdo it or else it becomes a snooze fest of rolling dice. Remember that complications can arise beyond just "you fail", such as weather problems (comes into play with the survival aspect earlier) or maybe even it gets taken over if the players leave it for adventures, which starts a whole new story arc. 

Sorry for the longwinded answer, hope it helped. 

To echo one of the other posters, this is D&D 5e.

If you don't want to use skill checks to evaluate the usage of character skills then what DO you want to use? To hit, skills and saves are the mechanics that D&D 5e makes available to the players and the DM for resolving situations where the outcome is uncertain. You can make the evaluation more or less complex, ranging from auto-success, one skill check, chains of multiple skill checks, automatic failure, but it always comes down to evaluating whether the character was successful in the task they were attempting.

I'd also add, that as you have discovered, traveling cross country with multiple random encounters is BORING, time consuming, not fun and will usually kill the enjoyment for most if not all of the players eventually since random encounters typically serve NO purpose. The DM may feel that having the random encounters makes the world feel "scary ... oooh so dangerous" but I have to ask, unless you actually want to have the possibility of killing the party off with these encounters then are they REALLY dangerous? Or do they just take up time?

Also, you mention it is a survival D&D game in a post apocalyptic world, but it is important to realize that survival by itself will not be a long term motivation for the players to keep playing. Survival by itself is also boring. At the end of the session, the players don't discuss the cool achievements they might have, they discuss, "Well we survived again, I wonder what comes next? More 'survival'?"

Survival can be a useful subplot to put pressure on the player/characters to achieve their goals quickly to minimize risks and increase chances of success but it usually won't sustain a game.

In your example, an apocalypse caused the monsters to go "insane" and attack everyone else.

"What caused the apocalypse?"

"Are the monsters still insane?"

"Is there something that can be done to restore the monsters to sanity and restore some balance to the world?" - This would allow humanoids to begin to reestablish some sort of society and civilization.

"Can the characters find a way to travel back in time and prevent the apocalypse? What information would they need to discover to make it possible? Are there other creatures about with the knowledge but lacking either the ability or resources to attempt it?"

If the entire point of the campaign is survival then it won't last unless the DM and players really just love having a random fight of the week, every week, without any point except surviving (and some folks really do enjoy that so if that is the case here then go for it - but otherwise, the campaign needs something more than just survival to keep going in the longer term).

 The answer, for me, is Skill Challenges. Here's a Matt Coleville YouTube vid about them (worth watching).

I twist it with the notion of roles from the Forbidden Lands RPG and go with this:

1. One character is the Navigator and makes a Survival check to determine how well or poorly they navigate the terrain.
   • Does the party get lost? Stuck at some kind of dead end? Travel as expected, go faster than expected, find a shortcut?
   • The Navigator's check determines the result.
2. One character is 'On Watch' and makes a Perception check to determine what, if anything they observe on the trail.
   • A good roll will help either avoid an ambush, or detect it before it happens.
   • A good roll might uncover secret locations and ruins.
   • A bad roll might result in being ambushed.
   • A bad roll might trigger a planned random encounter (vs. random).
3. One character is the Provisioner and makes a Survival check to provide sustenance for the party
   • Do they find good sources of clean water?
   • Do they find convenient food sources? How many people does it feed? Are the leftovers preserved?
4. One character is the Naturalist and makes a Nature check that assists the other rolls.
   • They get a read on the upcoming weather and allow the Navigator to better plan the path ahead for it.
   • They help identify what should and shouldn't be in the local for food and water sources improving the Provisioner's results.
   • They can identify unusual behaviors in the local animals helping the person On Watch anticipate things that are of concern.

This let's multiple characters get involved in the travel and flex their skills to improve the results wihtout adding new rules like crazy. As the DM you set a DC for the rolls and higher means they do better than hoped, lower means worse.

It's another topic entirely but I'll be quick on this as you can run with it:

There are no random encounters

(in my game)

I make a few encounters for a region; each feeds into a little quest or something. Encounters that are 100% random deplete party resources, but don't add to the big story. The encounters I make are structured like random encounters, but have a story connection to the locale. Why are these monsters there? What can they do to advance a story point? What can they do to add excitement for the characters?

Even a mindless beast is telling a story to some degree. This is, in essence, curating the random encounter table to make it a potential and planned encounter. If there's a goblin patrol they come from some base of operations and have wants, needs, and goals. So the random goblins should mean something. An owlbear in the same region might be feasting on goblin scouts and have a lot of their bones in its lair and immediate vicinity. Which then points back to that goblin base of operations, etc.

I ran a campaign that ... well, died, eventually ... but it was like this: You're an advance party for a clan of goblins. You arrive at a highland plateau, some 100x100 miles. You have 100 goblins, one ogre, 20 wolves, and limited supplies. What do you do?

What they did was build a base, start raising spiders and wolves, brew poisons, explore. They soon found a castle guarded by undead, an elven tomb, a human settlement, and so on. 

I free-formed most everything - which may have been a weakness - so I allowed them to breed stronger spiders and wolves, and so on. I also just informed them that 'this current base you have will provide food and shelter for half your number' - so they knew they'd need hunting parties, extra farmland, and so on. 

Oh, I just now remember they found a hunting lodge, with the long-dead corpses of the hunters tossed aside. That was the last encounter, and I think they just got to realising one of the hunters had turned werewolf, and shredded his own mates - and that they'd need a source of silver to defeat him.

But the basic setup was simple: Here's werewolf - go find silver. Here's vampire - you can ally with vampire against human settlement, or with human settlement against vampire (but neither means you any good, once you've solved their problem, which I didn't tell them of course). The main arch of the campaign was to eventually grow strong enough, working for one faction, to then defeat the other. Which would pacify the land, enabling immigration of the tribe.