What could be the reason why a Customer Service role is not able to see a shipping address when entering a sales order? Other roles can.

They don't have the permission on the role. Copy the URL for the address page, change to the role, paste the URL, and navigate to it. You should get an error message telling you what permission is needed.

The address is visible under the customer record but once they try to enter an order and dropdown in the shipping address, is not there.

It's very likely permissions; have you tried comparing permissions between the CS role and another where it's working to find which permission they're missing?