I was told that there is a bank in town that states that the Bank will not cover funds taken by a family member. Is such a statement be permitted?

Hard to say without more detail - is the family member a co-owner, agent, POA, authorized signer on the account in question?

Niether, it is in response to unauthorized access.

If it's unauthorized access, than it doesn't matter who accessed - family, neighbor, boss, employee - unauthorized is unauthorized and Reg E applies - period!

Now, the only thing that may have happened that I can think might cause a financial institution to make that comment is if the unauthorized access came about via an access card that the customer gave to the family member to make a purchase, along with their PIN. That situation could cause a bank to deny the claim.