Are you looking specifically for vegan shoes? Because if you go to a running store they won’t know about animal products and stuff like that (probably not – I haven’t had great experience), so you’ll want to do your homework beforehand and know which brands are vegan. Or look them up in store.
I really like Asics. I like the gel venture for street running, a a whole bunch of the trail shoes. Also, a lot of asics are pretty cheap! I have the Merrell body glove, which is a barefoot trail shoe that’s good. I have my eye on Brooks for some time now too