Actually, it's not that surprising. Whoever developed the application chose
to use references over and above the basic ones associated with Access.
Whenever you do that, you run the risk of incurring problems with the
References collection, as you are. The probability of problems increases
significantly if differently software gets installed.

The three controls you mention are actually part of the operating system,
although you technically need a Developer product (VB, Office Developer or
something else) in order to add them to an application.

Are you saying that opening the MDBs on the new machines works? If so,
simply create new versions of the MDEs. Your application should be split
into a front-end (containing the queries, forms, reports, macros and
modules), linked to a back-end (containing the tables and relationships).
Only the back-end should be on the server: each user should have his/her own
copy of the front-end, ideally on his/her hard drive. In that way, you
simply have to ensure that each user has whichever version of the front-end
works for his/her machine.