I live on the Westside, so for very good Oaxacan food, I recommend Monte Alban. My veggie sister likes, also. It’s not far from UCLA and they serve breakfast and are open non-stop until close at 10pm.
Also on the westside is some of the best sushi in Los Angeles!
My husband worked at UCLA for around 11 years and we used to enjoy occasional lunches in convenient Westwood at Skylight Gardens - at least one good craft beer on tap, good handrolled pastas and they would make me an off menu item - Caesar salad with deep fried squid. Their steaks and fries are good, also!
Also in Westwood in the pizza chain that can satisfy for most palates and diets, if you want something faster.
I’ve taken an Australian friend here who loved it, and my veggie sister. It’s not traditional Mexican, but it is incredible food and a great spot for lunch downtown.
I agree that the best things about LA is our ethnic cuisine, so I would seek out places that serve amazing regional cuisine.
I would make a beeline for dim sim at Longo Seafood in Rosemead. There are vegetarian options.
My favorite Thai place with an amazing Thai dessert place in same mini-mall is Ruen Pair, but I think vegetarians would have a hard time here.
My husband’s favorite Thai place is Jitlada, which I think would have more vegetarian options.
I also love Musso&Frank for a boozy lunch in the middle of hollywood.