Is an employer allowed to require employees to pay for business related expenses to be reimbursed later?

It is perfectly legal--and common--for employees to have to pay their own expenses and then apply for reimbursement. In fact, between my wife and I, for example, we have 60 years of work experience and we have NEVER had the employer pay travel or other business costs upfront--it has always been paid by us, then reimbursed.
As you in fact notice, there is no law requiring reimbursement at all--it would be legal for an employee to simply have to absorb some or all work or business expenses him- or herself. (Though in that case, the employee could, if he or she itemizes his/her taxes, likely take some or all of them as deductions.) Rather, the employer's obligation to reimburse is governed by contract law: if the agreement or understanding, whether written or oral (unwritten) between the employer and employee prior to expenses being incurred is that they would be reimbursed, the employer is contractually obligated to do so, and the employee could sue the employer for "breach of contract" if not reimbursed. (Note, however, that oral agreements can be changed by the employer at will on a forward looking, or not retroactive [i.e. not for expenses already incurred] basis, so long as the employer notifies the employee of the change.)
Requiriing emloyees to pay first, then seek reimbursement, does not violate minimum wage or any other labor law, since the employee is still being paid his/her wages. If they are not reimbursed, then based on the exact circumstances, could be a violation of the FLSA, since in that case, if they reduce earning below minimum, they would run afoul of the law.