Almost everybody is off topic. Please read the OP carefully. This really isn't about the routing number.
Oh, ok, gotcha, you are correct sir. Ive run into this, I try to get them to find at least an old check book that may have an old deposit slip in it, the information is on that, at least the name of the bank and account number usually are. If that does not work and they cant locate anything:
1. If they have a debit or cc write them with that carrier that will take those. I know its a pia, but its not your fault they cant find their own info.
2. Write with a carrier as direct bill - Your mileage may vary, I dont do it.
3. Leave.