Well, she serves Bishamonten, so I guess she is Buddhist by default. Then again Shou is Bishamonten's avatar but she drinks - a Buddhism no-no. So who knows, really... The rules of Buddhism in Gensokyo may be a little warped.
The Myourenji isn't the one and only place a person can practice Buddhism in Gensokyo. I'd say her not living there isn't a big deal in that regard. I'd be more concerned about her eating meat and inducing lustful thoughts in humans.