Technically, you might be able to get a more modern 32-bit browser to run on top of Win32s (which gives you a subset of 32-bit functionality for 16-bit Windows). If you want to try that, I'd start with "portable" versions of Opera, Mozilla/Firefox, etc. (I also mentioned this in a comment the other day, but deleted it because I didn't think it was advisable.)
That said, you might want to reconsider your decision not to try running the software on a newer machine. As Rich pointed out, there are many reasons not to run a web browser on your bandsaw controller PC.
However, at some point something bad will happen to this PC, and at that point you're going to lose a lot of money if you don't have a well-tested backup plan in place. I don't know much about lean manufacturing, but I do know that when a critical system goes down with no contingency plan in place, it has an immediate and direct effect on operations.
Something that won't cost you much right now (aside from time) would be to try copying the Windows 3.1 installation to a VM or emulator such as DOSBox, or even try to run the software directly on a newer version of Windows. Many programs won't run on the first try, but can be made to run with the right compatibility options. I was pleasantly surprised several years ago when I was able to make an insurance company's proprietary DOS app run just fine on Windows 2000 and XP by doing nothing more than supplying an extra flag to command.com or cmd.exe. (Note that command.com is apparently only available on 32-bit versions of Windows 7.) Even if you can't get any support from the manufacturer, you may be able to figure it out yourself or find a local independent computer shop or techie who is up to the challenge.
Also consider any processing time that you're currently wasting. Maybe opening and saving files for your bandsaw controller is instantaneous, but I wouldn't be surprised if you could recover at least several minutes a day by shortening or eliminating all the "loading..." screens. The insurance agents in my anecdote were ecstatic, because the obvious side effect of migrating to newer machines was that everything ran faster, so they were no longer waiting for screens to come up.
If all goes well on a test PC running a more modern OS, you can set it up as a permanent replacement and keep the old PC as a backup. (You should be able to pick up a 5- or 10-year-old industrial PC for pretty cheap, if a regular desktop machine isn't likely to survive very long in your environment.) You still probably don't want to run a web browser on it for real-time performance, security, and uptime reasons, but at least you won't have to worry as much about that fateful day when the PC breaks.