When I went to the lodging I had booked, I was told they didn’t have my reservation.

Even if you’re told that they don’t have the reservation, don’t give up. Talk with the manager of the lodging and show them your communications from when you made the reservation. If you have a record such as a screenshot of the reservation screen or a confirmation email, show that. If you reserved a private residence that takes lodgers (minpaku) through an online booking site, there may be rules set by the site for how they handle such trouble, so reconfirm the terms of use and make a claim to the booking site.