If the storeowners allow it, there's nothing stopping you

If the dog spoke English, I'd play in that game.

There's also nothing saying the dog is going to be any good at it, but I've seen a few humans do worse than I'd expect from a Chihuahua so I guess it would be better than nothing.

Who says you have to have an AL adventure? I've played in stores a ton but never used one that I recall. Nothing against them, some of them are pretty decent adventures, and they'd be a fine choice, but they'd only be necessary if someone insists that you use one.