How can one map IDs from two systems if one or both don't have the ability to specify the other ID?

Sometimes you might need to get creative.

For instance let’s say one of the systems allows you to “tag” records with a hash tag or something. Or even put that hash tag into a description field in the target system. You can put in tags like “#ID-ACCOUNT-1211212”.

Another trick is to use an external database. Just have a simple table which maps IDs from the two systems to one another.

There is always a solution.