To answer your questions, a recessional is technically not part of the liturgy, so music does not have to be “liturgical” to be used as a recessional. (Of course, it should be suitable to the dignity of the church and the occasion.) And there is no website of approved music. You should coordinate your musical choices with the music director of the parish where you are getting married.
As an additional point, I find the term phrase “liturgical music” so vague as to be nearly useless. In one sense, you might say that liturgical music is music written to be used during the liturgy, so that if the composer intended his music to be so used, that makes it liturgical per se. Or in another sense, you might say that liturgical music is any musical setting of a part of the liturgy; then purely instrumental music (and hymns, like “Holy God We Praise Thy Name”) can never qualify as liturgical. Or in a third sense you might say that liturgical music is music that is appropriate for the liturgy. Then the problem is the circularity of the definition and its reliance on judgment calls. Or maybe liturgical music is simply music that is used during the liturgy …
