I just found out that statements that cycle last week have not been mailed out because the stmt folding machine is broken (suppose to be repaired today). Reg E states that the bank shall send a stmt for each monthly cycle...but I can't find a specific delivery requirement (X number of days after the 30 day cycle) to quote to mgmt.

There is no such specific requirement in the regulation.

Where you may have issues in the mailing date is any long claims that involve the statement mailing date plus 60 days. So whomever handles claims needs to know that this cycle was mailed late. (Likely this won't be an issue, but just in case...)