The 39 Air Rifle matches fired under the National Council rules make allowance for collegiates and adults. Sporter class limits one to glove, sling, and normal street clothes.
Find a match being held in your area, and contact the match director to see if they will include an "open" category. We tried it and had three collegians one year, and a father who wanted to shoot with his daughter (but couldn't kame it at the last minute).
That would be sporter air rifle. Take a look at the rules to see if that's what you're looking for. In short, you can only wear 2 sweat shirts, you still get a glove, a sling, a mat. But no jacket, pants, expensive rifles...
There is also NRA Light Rifle for .22s. Check with the NRA Competitions website.
That said, there's nothing to stop you from competing in a regular air rifle match in just street clothes, if that's what you're comfortable with.