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 website througwebsites 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?