My main thing is that I don't know if I will stick around for end-game stuff which is why I was thinking of starting with vanilla. I mainly want to try out the story/ campaign and I know there isn't too much of it in the normal version but I'm worried about getting locked out of stuff relatively early in the game.
Would I need to hit the level cap before I can even start the DLC?
Not really the full cap. The max cap is L34. To start The Dark Below / DLC 1, you need to be 25. To start HoW, you need to be L28.
Vanilla can get you to Level 34, just slower since you have less options. Mostly need Etheric Light which can be obtain once a week from a Nightfall, Iron Banner which is once a month for a whole week. Then for the 2nd DLC, there are 2 more options: PoE and Trials, which are both end game contents.
The end game content is the best! Vault of Glass raid, you have to be Level 26, that's vanilla and a ton of fun. Crota's End is ok, the 2nd raid which comes from the 1st DLC. And if you have any interest in PvP at all, Trials of Osiris End Game from the 2nd DLC is A LOT of fun.