As far as I know, you need to use ODBC. Is it at all possible for you to recreate the database with MySQL? You could install Apache, MySQL, and PHP on that computer (Or use a server they already have setup I suppose?) It shouldn't be hard to recreate this database, PHP and MySQL just work so much better together than PHP and Access... Really you don't even need the Java... but if that's what you like to use then there's no problem.
As far as the program specifications...
As you said the checking system wont really work, as they can tell each other the info. All you (or anybody for that matter) can do is have a username/password and tell them not to tell anybody.
All of the other things are reasonable, at least with MySQL, not entirely sure with Access. Do you think they would let you use MySQL instead? If not, for the ODBC connection, you shouldn't need to do that on every computer, just on the server. Since PHP is server-side, you have the client computers access the page and when they submit, everything is handled on the server.
If you have any questions let me know, I will try to help. If you can get MySQL set up, that would be great, if not... Access is the bane of my existance