Is there a modern browser that runs on Windows 3.1?
Sorry to put this as an answer, since I can't give you one, but seemed too big for a comment.
To add to the people recommending not to do this: the more useful that you make the Win 3.1 machine (by allowing it to do other work), the longer the machine is just useful enough to not replace with something that makes sense.
Win3.1 is a horrible OS to have as a controller. Think of it as DOS with a GUI, which is what it is. There's no process isolation so a bad browser window (or any app) can scramble memory in the controller's memory space wreaking havoc. Since you have old browsers looking at modern HTML/CSS/JavaScript, there are a lot of possibilities for browser bugs and the odds of scrambled memory are relatively speaking, high.
There's also not a lot of Win3.1 support for realtime. Even Win95 was better, though not by much. This means that as apps try to cooperate and share the CPU, they don't necessarily have to, and some app or browser window may use up all of the CPU, throwing off the timing for the controller software. That may not end well. This is in fact a bandsaw.
There were (possibly still are) viruses for 3.1 (err, DOS really), so opening it up to any network should scare you. Any wise network engineer would require a firewall (if you could find one for Win3.1) and at that point you bog the machine down. Then, many answers recommend IE5, which is unpatchable at this point.
Win 3.1 was end-of-lifed about 4 years ago (surprisingly recently in my book). Internet Explorer 5.x had its last support of any kind in 2010. I'd tell my boss that this is connected to a bandsaw and we don't want any problems with the controller software. If the controller-software people were at all smart, they'd have a clause not allowing any other software on the controller box, or you void some support from them.
If you really need something close to the controller, I'd say WiFi+cheap tablet/used iPod Touch, but I'd obviously say you need to secure your WiFi network as well.
Opera 3.62 might be the best match to a modern browser, that still runs on such an old system like Windows 3.1 (which even has no native support for TCP/IP). HTML and CSS support are quite nice for the age of this browser, but don't expect too much. Also note, that Opera 3.62 has no support for dynamic changes in websites through manipulation of the DOM, which makes modern JavaScript driven web applications unusable.
I would suggest to use another computer for the time clock software. Apart from your problem such an old system should only be run isolated from networks, because there are known unpatched vulnerabilities. On the other hand, one might argue, that there is virtually no network spreading malware runnning on such old platforms. Anyway I would not risk using such a special machine to do anything else apart from controlling the saw. What would you do if it breaks due to using it for tasks you could do on any other computer?
Err... Do you really want to give a second task to a computer from the stone age that is controlling a $150,000 bandsaw..?
Judge to plaintiff: Tell us how you lost your arm.
Plaintiff: Well, Bob was late to work that day and tried to punch the clock while I was using the bandsaw.
Prosecutor to defendant (Bob): Did you know that the bandsaw always stops while the time clock web site is opening?
Defendant: Sure, everyone knows that. We'd just wait for the person to click the Clock In or Clock Out button and then the saw would come right back on!
Plantiff: I didn't know Bob was late! The saw stops when a part gets stuck. How was I supposed to know that it would suddenly start back up while I was looking for the part in there..?