Sfx104, there never was a company called Rockford Arms which actually made shotguns. In the past it was common for a retailer to order guns from a supplier and have their own in-house brand name stamped on it. These guns are called "hardware store guns". Rockford Arms is such a name.
Some hardware store names are famous, like Eatons or Sears. They moved thousands of guns with their own brand names on them. Others are less well known and some are very obscure indeed.
Either way, they all bought guns from existing manufacturers, some domestic and some imported. The lack of proof marks on your gun would indicate American manufacture (assuming that you have removed the forearm and barrel and looked for proof marks stamped on the bottom of the barrel). There were several American manufacturers who made guns for the hardware store trade.
If a picture or two of the gun were available then someone might recognize it as a particular model from a known manufacturer. The only other choice is to start searching libraries for old hardware store catalogues, looking for the Rockford Arms name. That could take some time.
Sharptail