With a FOIA requuest, you don't ask a government official, you ask a government agency. A government agency isn't protected by the 5th amendment.
In practice, the response to a request isn't provided by an agency, but by an employee of that agency. That employee should respond based on the knowledge of the agency. If the employee has any private knowledge, that wouldn't become part of the response. That applies if the employee has private knowledge of a crime. If that crime was committed by someone else, it might have been illegal not to report the crime, but that is independent of the FOIA request.
Now let's say the agency has knowledge of a crime that "the agency" committed. (In reality I would assume some member of the agency did). Since an agency is not protected by the 5th amendment, it has to be part of the response.
Now let's say the agency has knowledge of a crime that the employee writing the response committed. That's when self incrimination comes into play. I don't think 5th amendment allows you to lie, including lying by omission. So quite possibly that employee can say "I'm not going to write the response to the FOIA request". In that case, the next employee would have to write the response, and that employee wouldn't be incriminating himself.
Now if all employees committed a crime together, then they might all be able to refuse to write the reply, but the agency still has to respond, so they might have to request outside help :-)