The FDIC Statement of Policy https://www.fdic.gov/regulations/laws/rules/5000-1300.html [Moderator, please convert to a link] goes into considerable detail and probably could have been cited as answering your question if you had provided any information about the specific crime this individual committed.
Some personal observations:
1) I would have to be incredibly excited about a potential employee to go through this process vs. hiring someone who simply does not have a criminal record.
2) If I had any doubt about whether an application was necessary I would apply.
3) Before going through any of this, I would contact the issuer of my fidelity bond to see if they had an issue. (Any FDIC approval will be so conditioned anyway.)