Whether through physical abilities or bs-ing something, all characters will have the ability to do basic functions: breathe, talk, move about, etc. It's just anything extra that's getting nerfed—you can fly, but you can't fly at supersonic speeds, for example. A mouthless alien that communicates via telepathy could be able to use telepathy to hold a conversation but could not read minds, erase memories, do a psychic attack, have a telepathic rapport with one person in a crowded room, etc.

I haven't read Animorphs in literal years so I don't know the extent of thought-speak, but if it's used solely for general communication, then it's doable. Of course, you can also slap a mouth on the horse if you want.