Access can normally cope with a mere two users, so you may have an other problem here.
Though only a couple of tables with a narrow amount of fields, it is possible that any locks are opage based and therefore cover a lot of data.
Thing is if this was a concern, why did you pick access in the first place, it's notoriously weak in this area, it'sa single user power user analysis tool.
Personally I'd stick sql server express on your 'server' and link to it from access, it will either get rid or highlight your problems.
Failing that try a compact and repair (after backing up the mdb !), if you have an ongoing design/implementaion problem that's likely to be a te,orary fix , but you'd want to do before flopping to a proper database anyway.
Can't help you pinpoint the fault based on this much information (if ever to be honest), could be anywhere, or even everywhere.
Access is desktop database, it has no client server capability. It relies on the machine you locate the mdb to manage writes to the file, in that it's no different to using a text file as your datavase and hoping one user doesn't overwrite teh other's changes.
You can make it cope and I'm surprised it's this bad under an apparentky small demand scenario, but you are always going to be fighting to make a poor choice of tool work. Cthulu help you if you get another user or three for instance...