Travel cards work just like any other credit card – they have due dates that need to be met. The University, understandably, is reluctant to pay late fees (unless the responsibility is on their end), so make sure to submit your paperwork as soon as reasonably possible after the expense – preferably at least two weeks before the card payment is due, to allow for processing time.
You do not need to wait for your statement to come out before submitting your Travel and Expense Reimbursement request. Make sure to indicate on the request form that the expense was on your Travel Card; if you select “Out of pocket”, the payment will get sent to you personally, and the card will not be paid. Because late fees negatively impact your personal credit score, you should strive to avoid this situation as much as possible.