The problem is almost certainly permissions.
Remember that through a LAN you are using the directory SHARE to open the MDB with Access and in this case, the LAN connects you via FILE SERVER operations.
When you connect via a web front-end, you are using IE permissions, which are different than LAN/SHARE permissions.
You need your network admin to compare "effective" permissions. In the case where Open Exclusive occurs, the odds are that you don't have proper access to the LDB file or the ability to CREATE the LDB file via LAN. Either would have the effect of forcing an Open Exclusive because Access cannot arbitrate its own fine structure without the LDB file being fully available.
Just FYI (and for your network admin), you need the equivalent of the 'coarse' permission "MODIFY" on the folder and all the files therein. Or if you need to use the advanced permissions, everything EXCEPT Take Ownership, Set Permissions, and Set Attributes. Even then, I'm not sure about the last one. In general, the only ones you DON'T need are the fine-grained permissions that are different between MODIFY and FULL on the coarse-grained settings on the permssions tab. That goes for the folder as well as the files in it.
You specifically must be able to open (for write), create, and delete the LDB file in that folder. Plus all sorts of similar "wide-open" permissions on the LDB and associated files.