Is it critical for a database to have a relationship table/diagram or not?
You don't need a diagram for it to work, you need one to train the future developers!

However, if what you really have is a database with no actual formal PK/FK relationships set up (Which you can do when creating diagrams as well as through better ways such as SQL scripts), then that is a different story. That is something that will eventually cause your data to be inconsistent and is a real problem indeed to other users of the database.