Anything is possible, given time and money.
Outlook Web Access uses the live address book(s) and GAL (Global Address List) in Exchange, not the Offline Address Book.
If I understand your question and your situation correctly, in order to access your "on-site" contacts via OWA, the IT firm which maintains EXCHANGE at the main site would have to IMPORT your on-site contacts into Exchange and create an Address List under the GAL. This would be fully accessible and searchable in the OWA address search functionality.
While this isn't too complicated a process, it is not user updateable, in other words, the IT firm would have to do any Adds/Changes/Deletes to/from that list ongoing.
Also, since all of the imported addresses would then appear in the GAL which all other users and sites use, the IT firm would have to rethink the whole GAL and address list structure for the company -- again, not rocket science, but a much larger scope than the immediate question.