We recommend reviewing your lease agreements for the safe deposit boxes and notifying your customers as required in those agreements.
We are unaware of any law that directly addresses drilling a safe deposit box when the lessee is alive and is current on lease payments, which would be the case here. Whether and when you may drill in this situation depends largely on the language in your safe deposit lease agreements.
If the agreements are silent as to moving items in the case of a branch closure, we recommend documenting the notices you have provided (i.e., with copies of letters and records of phone calls). When moving the contents of the boxes, we recommend carefully following and documenting your procedures for drilling the boxes, removing the contents, and reinstalling them in an alternative location. For example, you might make and retain video recordings when drilling the boxes, in addition to having two employees present, to defend against any potential claims that items are missing.