What exactly the problem you have?
Just to verify your doubt: yes, the application is actaully run in local
computer, all components the app references must present in local computers.
It sounds that you did not split the Access app into two parts of front end
and back end. In this case, a *mdb file contains both data part (mainly,
tables) and application part (forms, reports, queries...). Although the file
is located on a server computer, when user runs it, the application part is
read into client computer and runs locally (that is why you need to install
Access on all clients computers), data part may or may not entirely be read
into local computer, depending how the tables/queries is designed/performed.
It becomes clearer if you split the app into front/back end: the fron app is
installed locally, so all referenced components are required to be present.
Note, it is alway good practice to split Access app into front/back end,
especially if you have more users than just 2 or 3. It makes backup easier,
reduce the chance of data corruption, and prevent your code from altering
(by making front end a *.mde file)...