We have a business customer with a stamp used for endorsing checks that begins with something to this effect "Pay to the Order of Bank name for deposit only into Business account number..."

We have never seen this before starting with "pay to the order of Bank name." Would this be acceptable, or should we order another stamp for this customer?

The reason you have never seen it before is because it is nonsense. Unless the check is made payable to the bank and it is going into a bank controlled account it is not an accurate endorsement stamp.

I realize I left out part of what the endorsement says...the business name is also mentioned, along with the business account number. So, it says "Pay to the Order of Bank name for deposit only into Business name and account number..."

Well that makes more sense but I'm still not a fan of the part that says Pay to the Order of "Bank".

I've seen similar endorsement stamps throughout my career- but it is somewhat antiquated.

I believe the bank name is there because they provide the stamp and it implies that the deposit will be made to the business account at that bank.

That's the standard format for rubber stamp indorsements that I saw during 30 years in banking. There are others, to be sure, but there's nothing strange about it. It's a special/restrictive indorsement that identifies the transferee and what the transferee is to do with the check.