I've heard one archer correct another and say, "we don't shoot arrows, we cast them." To me, "casting" sounds too un-archery related. Like others have said, responding to the question of "what sport do you play?" with "I shoot archery" is a bit like saying "I kick a soccer ball." "I shoot arrows" makes it seem like you just shoot (or arch, loose, cast, fling, etc) without thinking or effort. (Which ironically is what we strive to do..)
Archery is hard to put into a verb. How about responding to the question with a sentence about it instead? "I am an archer" or "I shoot- oops, that's not a noun for the sport.
Combining "shoot archery" into a single verb is hard. I wish it didn't sound too awkward. "I practice archery" is good, but sounds super formal. We could just say that we "play archery," because if it's fun, it's play!
I like this discussion - I was just thinking about this when writing a bio for the community theatre's program. I ended up putting "she enjoys... competitive archery."