RE: All of the Mail data has disappeared from the admin screens, and the forms do not function.
Something has definately gone wrong.
RE: I've checked in the database and all the Mail data is there for each form.
Good - also means we don't need to consider that issue.
RE: The client had activated five new plugins.
Sounds like most likely cause of problem. Did client take back prior to doing this?
RE: tried deactivating all of them, but no change.
Deactivating a plugin does not always remove all problems.
E.G. Known Possible Conflicts
WP Super Cache - Apparently the WP Super Cache plugin is not fully removed when deactivated and creates an ongoing error.