We know that your shop is running on a tight budget, so I would upgrade everything to something new and eventually minimize future needs to update - but your description doesn't tell us some things we need to know in order to best advise you.
1. How many distinct machines are you using? Access license costs are actually not per user, but per desktop. I.e. if you have a case with a machine used on more than one shift, but only one person ever logs in at a time, that counts as one user. Whereas if you used something like a Windows Server (possibly with a Terminal Server option) and have multiple users simultaneously running Access on the same server, that is many users at once for one copy of Access - which is where per-user expenses come into play and those license fees can start to build up.
2. Are any of your users running on machines that DO NOT belong to your organization? I.e. are some of your clients running on machines not under your control - and thus cannot be upgraded? That can be a really big problem to coordinate the changes.
3. What versions of Windows are we talking about here? Win11? Win10? Earlier than Win10? Mixed bag of versions? Upgrading is still possible but with limitations for earlier versions of Windows.
As to this statement:
Which Microsoft version of Access do you suggest that they upgrade to?
Access 2007, 2010 or 2013?
That is going to be difficult to answer in one way and simple in another way. First, we would recommend - if possible - to get the newest version of Access. CJ's suggestion is interesting. But that only works well if you have a uniform version of Windows among all your users.
Here's the good news: If you don't change designs on anything, your users might not see much difference at all, because the "look-and-feel" follows the DB's internal design choices, not changes in Windows or Access. If you don't change the design elements, it should be VERY CLOSE if not identical to prior behavior. (Exception: IF your databases use Windows Theme options for coloring controls, forms, and backgrounds, new versions of Windows have changed themes.)
Here's the bad news: Among the various changes that have been made to versions of Access over the years, one of them is that some features have been found to be insecure or unreliable, and therefore, certain behaviors HAVE changed as an intent to improve the overall product. I cannot tell you where to look because those changes have occurred over a span of 22 years and some of them were quite subtle.
There is also the matter that it might be difficult to upgrade because a database internal format change occurred between Ac2003 and Ac2007 that might make things difficult. In fact, the Ac2003 DB MIGHT be hanging up on issues related to that change that finally caught up with you. There is no way for us to know and I'm not even sure if I could remotely diagnose that issue (not that I'm offering to do so.) But if you COULD do this, you would eventually want to upgrade the DBs to the .ACCDB format rather than the .MDB format, particularly if you decide to go with the Office 365 package.
If you have now, or could get, a single copy of Ac2007, you would be able to convert each .MDB file to the .ACCDB format. Basically, open the .MDB file (written by Ac2003) using Ac2007, but then do a Save As to the .ACCDB format. If the modules compiled correctly in Ac2003, they should probably be OK in Ac2007 - unless one of those security changes applies to something you were doing.
NOTES FOR PLANNING PURPOSES:
1. This change to .ACCDB is an irreversible change in that once you update the format to .ACCDB and update to use the newer libraries that go with it, you will not be able to use Ac2003 on those converted database ever again. Access automatically upgrades to a new version but NEVER automatically downgrades back to an old version. Therefore, ALWAYS ALWAYS ALWAYS make a backup copy of the old DB before upgrading. That is your only path back for a failed upgrade.
2. Since the databases were developed on machines that didn't include the 64-bit option, ANY upgrade of Access will have use the 32-bit version. This doesn't mean that someone on a 64-bit machine cannot run Access. It is that if someone has a 64-bit version of Access, they cannot run a DB that was developed in a 32-bit Access environment. Running 32-bit Access on a 64-bit Win11 machine works perfectly. (I'm doing it.)
3. Your goal HAS to be to get everyone upgraded to the same (but newer) version of Access. From note #1 above, remember that Access will automatically upgrade if it can. If you upgrade to Ac2010 but a couple of folks have Ac2013 (as an example), the moment that the Ac2013 user touches that Ac2010 DB, it gets upgraded and the Ac2010 user will be shut out. So your plan HAS to reach uniform versions to keep from whacking the users who haven't upgraded yet.
