Is there nothing on it that indicates a brand? Is it old? How old? Do you
know who built the house? If they're still in business then ask them what
locks they used.

If there are other houses in the neighborhood that used the same hardware
then ask around for ones left over from remodeling efforts. I managed to
pickup a number of spare handles and doors for another house when a neighbor
decided to renovate.

Given the odd size of that assembly it's unlikely you'll find something that
will fit it perfectly. With luck you might find something close. But if
not it's really not all that hard for a decent woodworker to refit something
new. The door would probably end up needing to be repainted but the job
could be done well enough to look good. Probably trigger repainting the
nearby trim and room though...