I looked into this just a little bit in connection with an organization I was involved with a few years back. There are Internet companies that do basic screening - a public records check for criminal convictions and offender registry checks - for substantially less. But I'm not sure how to establish that a given service is legit.
The Amateur Athletic Union, of which the USFSA is an "allied member", uses the NCSI Check It Twice service, which cost at least $20/individual, and much more for foreign nationals.
I'm not sure if USFSA clubs have to follow AAU rules and recommendations.
If they do, I don't remember the exact rules and recommendations, but I think they required all adults in an organization (not just those with contact with minors), as well as children in positions of leadership, to be screened. There were a host of other rules, like that you could not have just one adult in a room or other "secluded contact" with minor(s). And you couldn't drive kids anywhere (or allow carpooling of kids with different parents) without written parental authorization. I'm not sure what was recommendation and what was requirement, and both may have changed by now.
Many (perhaps most) states, county and municipal governments have additional laws. E.g., some require checks by their approved organizations, fingerprint, have mandatory reporting requirements for a number of things, require specific certified training, etc. A good place to start would be to contact your local county and municipal government offices. Any answer to these kinds of question won't apply U.S.-wide, let alone world-wide.
Presumably a lawyer would make additional recommendations for liability reduction.