How to run Microsoft Access on a Mac?
Unfortunately you have found the conundrum Microsoft has placed before you. There is no way to run Access natively on a Macintosh. You can try Parallels or Bootcamp to run Windows on the Mac, and then within Windows, Office Access, but that seems a little like a Frankenstein solution.

I know this sounds heretical from the completely Mac-centric IconDaemon, but sometimes Microsoft wins and you have to bite the bullet and run a PC.

Have you considered purchasing access (pun intended) to a co-located Windows server? This company offers such a service. (No financial or other ties to Colocation America - I just Googled and this was the first company to pop up.)