Would vinyl plank flooring allow liquids to leak through?
Vinyl plank will allow the urine to seep underneath, unless you use a sealant on the joints/seams. Even if it is properly installed.

Using sheet vinyl will work perfect for your application. The only way urine or any other liquid could get underneath would be if a hole/tear was made into the vinyl from mistreatment or from a big spill seeping in from under the walls.

Also you can get the room floored cheaper. You can also get some sheet vinyl with wood grain in any tone/shade to match the room decor. If you do need a seam in the room a seam sealer is used - like epoxy, so no leaks

I've been installing for a long time and I have dogs that have accidents. I have sheet vinyl down in their room (12'x18'). No leak throughs. Easy cleanup, no joint lines to scrub.