The rules for being surprised doesn't consider that someone can be surprised in mid-combat. How to handle this? I will give an example.

Players are fighting Vine Blights, which have "False appearance" and cannot be distinguished from a normal tree. 

Ok, combat begins, for example 2 players vs. 2 vine blights. Somewhere in mid-combat one player get closer from another vine blight, who stands up and attack the player. The attack can take place immediately? Is this player surprised since he didn't "saw it comming"?

There is no surprise in the situation you describe. No character who is actively in a combat can be surprised.

You should roll initiative for all the vine blights so you know when they can actually act in the turn order. Because of false appearance, it's probably hidden, so it'll have advantage on attack rolls until it reveals itself.

Agreed, I would have everyone (including "inert" monsters) roll initiative.  The "inert ones, not yet actively involved in the fight simply wait on their turn, OR Ready an attack for anyone who comes near.  Same rules as everyone else and then the initial attack takes place (uses creature's reaction to attack on the trigger) and nothing more from it until it's proper turn in the next round of initiative.

Quote from Licozo >>

Is this player surprised since he didn't "saw it comming"?

 The attack could be at advantage (if the monster can make a stealth check) but is not otherwise different from any other attack.

Remember that Surprised is for when a creature is surprised about combat starting, not surprised by a new combatant.

As others have said, the hiding blights would have a readied action for when someone goes near them, and then would attack as soon as their move brought them into range. If the PC was trying to walk past, they will have to decide whether to keep moving or not, and give the blights an attack of opportunity.

Since there is no surprise once a combat has started, in such scenario advantage to attack or disadvantage to saving throw is something a DM could easily do as a result of having been unaware of it.

I have a different scenario. Party is fighting a lone Goblin who is tactfully retreating to an ambush set up yards away across terrain. Once the party is lured close enough the ambushers attack. Now the ambushers weren't near enough to the initiation of Part v. 1Goblin to have to roll a secret initiative and they are stealthed and would "surprise" the party with their ambush. How do you work out the mechanics of it?

Quote from IScreamsalad >>

 How do you work out the mechanics of it?

Roll Initiative for the goblin ambushers, either at the start of the combat or on the turn when they get to act.

Make a contested check - the ambushers' stealth vs the party members' perception. When the goblins attack, they get advantage to attack any party members who did not notice them, and they attack the rest of the party normally.

My thought would be the ambushers (if not perceived) take their action immediately at the time of 'ambush' interrupting the turn order, with advantage(this is the surprise in my mind). They then role initiative and join the turn order.

I believe the proper method for any of these scenarios has been already offered.  Any creatures who will be involved with the combat should roll for their initiative at the start.  Any who are hidden and/or unseen, can simply hold an action for anyone who comes near, for example, or simply pass their turn when it comes up.  When their initiative comes up each round, the DM can decide of they act or remain hidden.  When they act, or attack, they would have advantage on anyone who hadn't noticed them (I would personally be using Passive Perception VS Stealth) and regular attack on those who had.

I have done this in several combats, with hidden enemies simply passing on their turn, or ready an action to attack the first foe that comes within striking distance, and it works quite well, for me.  It also, I am pretty sure, lines up with how the RaW would play it all out.

Quote from IScreamsalad >>

My thought would be the ambushers (if not perceived) take their action immediately at the time of 'ambush' interrupting the turn order, with advantage(this is the surprise in my mind). They then role initiative and join the turn order.

 That's only in your mind though, as written Surprise rules don't grant the ambusher any additional action or advantage whatsoever, it only affect surprised creature's ability to move, act or react until after it's first turn.

As the DM if you want to implement that mechanic  you can.  However once the genie is out of the bottle the players seeing that its a valid tactic, they will use it.

The "ambushers" would roll initiative as normal. Keep in mind that there is NO reason for the players to be aware of initiatives of creatures that have not appeared in the combat.

There is no "surprise" involved, the PCs are already involved in an encounter and are aware of opponents.

The "ambushers" would roll stealth. This would be compared to the passive perception of the PCs. Any PCs with a passive perception higher than the stealth roll of the hidden goblins would notice those specific goblins when they get within whatever the DM considers a reasonable distance to notice them. The DM could make separate stealth rolls for each hidden creature if they want and each PC might notice some, all or none of the hidden goblins.

A goblin that is hidden from their target will have advantage on their first attack as an unseen attacker. (That is the extent of any "surprise" benefit).

Finally, as mentioned above, the "ambusher" goblins could take the Ready action on each of their turns to make a ranged attack when the PCs get within a certain distance. This way, all of the goblins would get one attack when the Readied action is triggered, some of which would be subject to advantage if the goblin is unseen by their target. After that, the goblins would continue to take their turns when their initiative comes up. Given the way 5e works in turns this could result in the first PC that runs after the goblin turning into a pincushion as all of the goblins use readied actions to shoot the first target within range ... but that is how it would work.

Quote from David42 >>

The "ambushers" would roll initiative as normal. Keep in mind that there is NO reason for the players to be aware of initiatives of creatures that have not appeared in the combat.

 

 While you are right, nothing make players more suspicious than rolling initiative al alone or for no apparent cause : )

I find Roll20 handles that nicely, monsters on GM layer aren't visible to payers on the map or the initiative.