Is there a way to do this as a regular Joe who had a simple background check done?
Apparently not...
It is important to note that this retention and searching of civil fingerprints is authorized only for those individuals whose employment, license, or other benefit requires that the individual not commit a prohibited criminal action. Moreover, these individuals are provided with a Privacy Act statement and other actual notices regarding the retention and searching of their fingerprints.
[...]
...NGI retains the civil fingerprints after completion of the noncriminal justice background check. The fingerprints are retained regardless of whether there is any match to criminal history information. When the civil fingerprints are submitted for retention in NGI, the fingerprints are searched against the existing civil, criminal, and unsolved latent files. Latent fingerprints are fingerprints collected from locations or property associated with criminal or national security investigations. Likewise, once civil fingerprints are retained in NGI, all incoming civil and criminal fingerprints will cascade against those fingerprints, and latent fingerprint contributors may choose to have their latent fingerprints cascade as well. NGI will remove the retained civil fingerprints should the submitting agency request removal, or removal is required by court order.
This retention and searching of the civil fingerprints provides, in effect, an “ongoing” background check that permits employers, licensors, and other authorized entities to learn of criminal conduct by a trusted individual, unless the contributor does not subscribe to a rap back service. It eliminates the need for periodic rescreening of the individual and the resubmission of fingerprints.