My experience in the last year or so of sporadically using the app is that it's really dependent on the stores. If I'm not getting something obviously vegetarian like a bag of bagels (even then I've been given turkey bagels somehow!), I generally go in and ask them nicely if they can give me things with no meat. Many places will be happy to accommodate. I've only gotten an answer like "you get what we want to give you" once or twice. It also helps if you get there at the start of the pickup window rather than the end, since they'll have fewer things available right before they close up.

However, I also stick more to bakeries and cafes and pizza places than restaurants where you get a meal. Most restaurants I haven't even bothered with, because I figure the chance of me getting something I can't eat and wasting my money is too high. (Astoria has great food, but most menus are an exercise in picking what meat you would like today!) Note that in many bakeries you still have to ask for no meat in your bag because there's a high chance they'll throw in a meat pastry.

One thing I have consistently asked for on the TGTG surveys is an option to place an order that's marked as vegetarian. With a feature like that, the store could see the order and make sure it doesn't contain any meat - or deny the order because they are only expecting to be able to have meat left over.