Hi. You don't really need to get into MS Office at all.
The majority of current Accounting software actually provides multiple computer setups & access. I worked at a Plumbing / heating / etc company where they used Quickbooks Pro across an average of 6 computers in the office plus a remote set of connections from Home.
Look into other brands like Peachtree, and several others, or start looking into small-time software houses also for a more homemade (Reasonably priced) selection too, but the things You need have been done for decades.
If You don't need to re-invent the wheel, You'll find it's WELL WORTH a few hundred bucks to get something that's already had all the bugs worked out of it.
Otherwise, I highly recommend just creating a main webpage with a company logo & some file links to Your main database that You can use a simple netbook or tablet to connect to through the building's WiFi by IP address. Add a password if necessary, unless the files are password locked anyway.