A mother and daughter had a joint safe deposit box. The mother passed away. Am I to understand correctly that the daughter must send notice to the state of the proposed entry into the box. Once she has done that she may inventory the box on her own, without a bank representative?

Yes, A bank representative is no longer required when an inventory is taken.

Sorry Kristen. I didn't answer the 1st part of your question:

From the PA Dept of Revenue, 4/2012

No one is allowed to enter a safe deposit box, not
even a joint owner, except to remove a will and/or
burial instructions, which must be removed in the
presence of a bank employee. The bank employee
must complete PA Form REV-487 (Entry Into A Safe
Deposit Box To Remove A Will Or Cemetery Deed) to
record the entry and mail it to the PA Department of
Revenue. All other contents of safe deposit boxes must be
inventoried before they can be removed.
If the decedent and a surviving joint owner owned the box, the surviving owner should be present.
An estate representative must provide written notice
to the department at least seven days in advance of
a safe deposit box inventory.

So the daughter would notify the state, she needs to bring us a copy of the letter she sent along with proof it was delivered. Then she can enter the box and inventory the contents on her own correct?

correct

If the box is rented to a business, do we have to follow these procedures or can the other owner of the business get into the box?

The business would be the owner. We would allow the other owner to have access to the box and we would require a new contract removing the deceased owner and adding any new authorized personnel to the resolution.