The club should be consulting an attorney about this, with the expectation that whatever course they take, the club will someday have its procedure come under attack by some personal injury lawyer.
It may not be legal to have someone 'informally' access those records for this purpose. Plus, "charged" is not the same as "convicted".
Then you get into the judgement calls. Convictions how long ago? For what? What were the facts in the L.E. report, vs. the charges eventually filed.
What if you don't do this for all shooters, just potential members? And a non-shooter does something wrong at a trap shoot & hurts someone?
Or say an applicant fails the background check, does that mean he can't shoot there anymore, but the guys (non-members) that weren't checked can, though later are discovered when SHTF to have previously unknown skeletons in the public records?
Lots of issues here.