No. You cannot do a level 2 check on yourself nor can you do a level 2 background check online.
This is because the fingerprint to be used for the check has to be submitted and processed via the state repository. Individuals don’t have access to this. Also, the results of the screening can only be obtained by an approved government entity. This entity must also agree to the FBI’s terms.
However, you can still do a level 1 background check on yourself.
This is an easy process and can give you access to court records, address history, standard criminal records, and other personal information. Individuals can easily run background checks on themselves via some free or paid background check services. These companies have access to extensive public records.