Is it possible to make a character who masters every single skill, perhaps using the Legendary skill option? On Vanilla.

Yes. Using Legendary Skills, you'll need to be a a level 252 (I think, someone can correct me on that). That level will allow you to have all the perks in the skill trees.

So is there no level cap in vanilla?

There was a cap once before the final patch, but now there isn't a cap. Well, there is, but it's like 55335, a number no one will ever reach.