Generally speaking, I cannot imagine why it would be illegal to keep a record of payments made by photocopying the checks. You of course have the option of paying by cash.
In general this is just their method of doing business and keeping records, or so it sounds. Unless your state has some law preventing a person to whom you send a check for payment from keeping a copy there may be nothing wrong with their practice.
I am not licensed to practice law in SD, only FL so you will need to ask a SD-licensed lawyer this question.