Would vinyl plank flooring allow liquids to leak through?

You can add an underlayment with vapour barrier, or just 6mil plastic vapour barrier underneath.

Will protect the subfloor (wood) but not the vinyl seems.

For occasional "accidents" this could work, and protect the subfloor from long term soaking.

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.

Neither offer a completely impregnable barrier but locking laminate is particularly susceptible to water damage. Thermal fused sheet vinyl is often what they use in hospitals to create a surface that has no seems where liquids can seep through, but it takes special tools to install. Heavy gauge plank vinyl when properly installed is highly resistant to spills and odors (my mother has had it in her hair salon for five plus years with no problems) and it cleans up beautifully.