Is it legal for a property management company to require a convenience fee for every type of payment they accept, or must they accept some form of payment that costs nothing extra? It seems to me that the lease stipulates a contracted total amount of monthly payments, and requiring an additional fee to pay it effectively increases the rent.
Until it was found to be unconstitutional, Florida had a law against surcharges for using a credit card (the statute is still on the books, however). That was the only law against "convenience fees" related to making payments.
Whether or not a business will accept a particular form of payment (check, credit card, money order, traveler's check, cash) is up to business. Apart from credit cards, there has been no law against charging for accepting a particular form of payment, but that charge would have to be part of the contract – the lease would have to specify in advance what the processing fee is for money orders vs. cash vs. credit card. The residential tenancies law of Florida does not prohibit incorporating fees into the terms of a lease (as some states do), so the lease can specify "$1500 for rent plus $10 for payment-processing, every month".