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

In the US, the answer to this question is almost always YES.  However... a handful of hospitals have signed exclusive contracts requiring their patients to use certain family banks, so it is best to check in advance. 

Those hospitals who have made exclusive partnerships 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.