It seems shady, but it's pretty common for people working with kids or elderly or disabled to be fingerprinted. There's the possibility of abuse. Volunteers have a weaker affiliation with an organization than employees so some organizations feel the need to go a little further.
And I can sympathize with that. But those prints should be destroyed -- with a verifiable chain of evidence witnessing their destruction -- immediately after the background check. And people should have the right to inquire about whether prints submitted for background checks of this form are retained. And LEO organizations, including the FBI, shouldn't be allowed to lie.
More systemically, routine background checks on people who are not suspected criminals should be handled by an independent organization outside of the FBI.
They're not destroyed because the prints aren't just used to determine if the person has committed a crime in the past; they're used in case a crime is committed in the future, and the only evidence for the criminal's identity is fingerprints collected from the scene. Without the records, every potential suspect would have to be fingerprinted again. I'd rather have my prints on file and be automatically cleared of suspicion than have the police show up at my door to bring me in for prints and questioning, especially if there is a high-profile crime (involving children, for example) and the media are digging around the lives of every suspect.
If it's a background check, it's a background check. Otherwise it's "mandatory registration in the government's database", not a background check.
There's a difference between retaining the fingerprints of convicted criminals as part of their case file and pre-emptively collecting fingerprints on people who are not even suspected of a crime.
If there's anything less than a 0% false-positive rate, having a massive database of fingerprints to scan without any criterion will just make it more likely that the police will show up at your door because "your prints were found at the scene" of some crime.
Then why not just fingerprint everyone at birth (or shortly there-after)? Because that would be an blatantly unconstitutional violation of privacy.
Disincentivizing people from volunteering at a school by forcing them to surrender biometric data to the FBI for inclusion in a database that is normally only used to catalog information about convicted criminals is insane.
Then they should also have your complete web history too.
It will be helpful just in case there's a high-profile crime (involving children, for example), so they can check and confirm you're not a potential suspect. That way, they don't have to show up at your door or bring you in for questioning.