Has any one ever made a giant spider or large wolf spider a steed for a pc in d&d? And if so how did you do so without breaking the game?

In early beginnings 30 years ago, we had players riding on spiders for fun, and in Pathfinder Runelords there are goblins riding on worgs.

Not sure if I would allow it for players nowadays. Spiders would be slower, but could climb obstacles more easily. Spiders would be hard to control, and as their diet consists of humanoid food, one has to be cautious not to end up as a snack, and always provide masses of food, besides carrying it along.

I would allow it as a DM so long as it fit the narrative.   Introduce some animal handling checks every once in a while to make sure the character keeps control over the creature.  

I would and have allowed it, but riding a dangerous mount requires regular animal handling checks unless the character has the mounted combatant feat, in which case I believe that the character is better trained and more capable of controlling an exotic beast that has been trained as a mount.

They are medium (giant wolf spider) and large (giant spider... same size category as a horse by the way) beasts. I see no reason why they couldn't be used as mounts given the right lore and story to explain "why" one of these creatures has been trained as a mount.

It's actually a recommended use of an animal companion for a small PC. I would double (at least) the cost of the mount, and make them buy an exotic saddle. If they can justify the existence and can afford it, why not?

There's already some precedent:

Duergar ride a half bug half spider thing called a Steeder.  Duergar, Drow and Deep Gnomes also ride giant lizards.

https://www.dndbeyond.com/monsters/duergar-kavalrachni

https://www.dndbeyond.com/monsters/female-steeder

https://www.dndbeyond.com/monsters/giant-lizard

Using those as baselines, you can modify to suit your campaign.

I let players have custom mounts. All I do is homebrew it by making sure none of the stats are better than a Riding Horse or Warhorse.

Those are already my starting choices. I usually use the monsters base speed and if they have a climbing or stuff like that those stay.

Then I remove multi-attack and change the damage to be in line with one of the two above.

A small PC could have a giant wolf spider as their companion.

I think availability would have to be considered as well...

Where would one purchase a giant wolf spider that had been trained as a mount? I like the concept (!), but there should be some pretty substantial backstory imo.

Yes I have and honestly it worked out by simply rescaling my combat encounters to the new mount. But this really depends on what level your players are. If they are low level find more combatants that would be far more preoccupied with the spider than the squishy players. If they are high level make the player have a more advantageous position by simply staying on the mount than using it as a combatant and either make it the sponge for damage or a mount that can make them unhittable by melee combatants. But, in my opinion there is no way of breaking the game simply if something is too strong then you up the power of the things they fight and vice versa. Never be limited in your creative choices because you believe that the game was delicately designed and you need to abide by its rules. The truth is that the game is roughly hewn and it is really up to us as DMs to make it way more refined through our choices.