Will my delivery hospital allow me to use any family cord blood bank?

Usually the answer is yes.  However... some hospitals have signed exclusive contracts requiring their patients to use certain family cord blood banks, so it is best to check in advance. 

Those hospitals that have exclusive partnerships with certain family cord blood banks will argue that it improves their level of care, because their staff train with and stock the collection kits for the banks they have approved.  The hospital probably has a financial incentive too.  However, for parents it means a lack of consumer choice.