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

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.