Chicken powder is probably the biggest worry, and at some (many) places, it will be ubiquitous (vs. regular MSG).
Lunasia has well-marked vegetarian items, some may have (non-vegetarian) oyster sauce etc., so ask. Lunasia has had some ebb and flow of quality over the years. I think the Pasadena one was not great (relative to the one on Atlantic) the time or two I went to that one, but it’s been a while.
I go to King Hua quite a bit, they have the Chaozhou style ‘fen guo’ which are marked as vegetarian, and which I’ve always been told are vegetarian. They also have a taro paste item with tofu skin on the outside, also claimed to be vegetarian. The last time I was at Sea Harbour (ages ago), I was told there was chicken powder in everything. IIRC, Elite should have one or two items as well.
In general, some of the sweet items are likely to be vegetarian, if not vegan (lotus buns, honeycomb cake, etc.). Most places have told me that they do not use lard (for cost reasons) in the egg tart crust, so I’m guessing usually veg shortening or maybe butter if you’re lucky? Most places will also be willing to prepare noodle dishes to order, so if you’re going along with friends / family and just want to not starve, you might want to get them to do a stir-fried noodle dish vegetarian.
Mustard greens or Chinese broccoli w/o the oyster sauce should be Ok, I think most places poach in water rather than stock, though correct me if I’m wrong. Plain chang fen (cheung fun) will be Ok, however, the sweetened soy sauce that comes with it could contain non-vegetarian ingredients like chicken powder or oyster sauce.
One of the challenges, esp. on weekends, is that it tends to be super busy, so you’re not likely to get super helpful answers. Going on the weekday and talking to the manager directly may be a better bet if you want to get better answers. My limited understanding is that some Chinese Buddhist vegetarians eat oysters, so that may be one reason that you’ll see dishes with (real, not vegetarian) oyster sauce specifically labeled as “vegetarian”.
I live near Fine Garden and go there occasionally, but I think their dim sum is not very good, even by vegetarian dim sum standards. It’s admittedly been years, but I remember Vegetarian Dim Sum House in NY being decent.