VaultWarden is already it's own thing code base wise, so, it likely won't effect the hosted portion of it at all.
The apps though, sounds like right now, it sounds like it won't make a difference, but, I think moving forwards it'd make sense to consider forking and making VW branded versions.
With that said, right now, the only reason you can choose your own domain is because BW allows you to self host their own product and VW just made their API respond the same way as BWs, so, as long as BW keeps allowing the self host option, I would imagine that there really isn't much they can do about VW other than starting to play games with their API, which would also impact their paying customers.