The provision is so vague, there has to be some official procedure to determine if a person participated in an insurrection. That is why it gives Congress the power to make legislation determining the procedures to be followed.
I don't think it has to be a criminal conviction. I believe most of the people "charged" under the act were removed from Congress under the power of each house to remove members by a 2/3 vote for being in the Confederacy. It didn't involve criminal conviction but an official body did determine if they person was in an insurrection.