Are you familiar with refactoring? It's difficult to give specific advice because it depends so critically on the local landscape, but in general, factor out a layer and give it a good healthy coating of unit tests, and repeat until done. It is more complicated than I make it sound, sure, but in some sense it is also really that easy. Factoring out the data access layer sounds like a first step, creating some sort of service that actually unifies the data access patterns and then moving up from there, but I can't guarantee that.
All but the truly worst worst scenarios are better met with this approach than a true rewrite. In this case by "rewrite" I mean the creation of a new system next to the existing system that doesn't work until all (or at least most) of the new pieces are in place. If you've truly got an epic snarl, rewrite may be the only option, but odds are you don't actually have that epic of a snarl. With the proper approaches and tons of unit tests, a refactoring is like a rewrite in the end, except you have a running system the whole time. It can actually be slightly slower in total, but you also get the value of being able to choose when to stop and a continually improving system that is always actually running; it's only slower vs. a rewrite that actually succeeds and completes and that is not a sure thing!
With a database backing you do also have the option of trying to bring up a new system that also hits the old databases, but that will in practice require the first step I laid out anyhow, the refactoring of the data access layer, and once you've done that the use of the rewrite value goes down a lot.
Refactoring only goes so far in this case, since we can only realistically refactor one application. Unit tests help the refactored application but they don't guarantee that all of the other pieces (DB stored procs, MS Access, classic ASP, assorted other bits) still work.
All but the truly worst worst scenarios are better met with this approach than a true rewrite. In this case by "rewrite" I mean the creation of a new system next to the existing system that doesn't work until all (or at least most) of the new pieces are in place. If you've truly got an epic snarl, rewrite may be the only option, but odds are you don't actually have that epic of a snarl. With the proper approaches and tons of unit tests, a refactoring is like a rewrite in the end, except you have a running system the whole time. It can actually be slightly slower in total, but you also get the value of being able to choose when to stop and a continually improving system that is always actually running; it's only slower vs. a rewrite that actually succeeds and completes and that is not a sure thing!
With a database backing you do also have the option of trying to bring up a new system that also hits the old databases, but that will in practice require the first step I laid out anyhow, the refactoring of the data access layer, and once you've done that the use of the rewrite value goes down a lot.