Question about running multiple versions of Java

Currently at the company I work for we use an older version of Java for some of our payroll/operations (specifically version 7u67). To my understanding they all use internet explorer to access the page/program that allows them to do what they do (I believe they use Oracle, though I am currently unsure of what version). We all use Windows 7 Professional, 32 bit, and I'm assuming that for something like this PC specs wont be necessary, though I can provide them.

We've recently been looking into how to run 2 versions of Java side by side, which by itself doesn't seem too difficult, the issue is that they don't want to have to switch between versions every time they need to access the old software and then back when they use the newer stuff. I've been looking around for a while to see if it was possible to use the older version only for specific web pages or programs, but after coming up empty handed 3 hours later I'm starting to think it's just not possible to do it that way.

There's been a rumor going around my work that there was once a fix that allowed for this, but it seems to be lost to time, and I'm starting to think it's just a myth. I've seen suggestions online of creating batch jobs that switch the Path back and forth between the versions, but nothing resembling what these co-workers are suggesting can be done, and batches are a bit beyond the scope of my understanding.

I would really appreciate any ideas or suggestions, but this is kind of a last ditch effort as I'm assuming that this is a bit of a wild goose chase.

Is that a normal desktop program? Or a website with a Java applet running client-side?

Normally, Java is very good at backward compatibility and newer runtimes will frequently run older programs just fine. Have you identified a specific case where it breaks, just because of a newer Java version?
Asking the developer to fix the bug is of course the best way of solving that.

That said, a desktop program is easy enough to run with a specific runtime. Portable versions of Java are best for this (unofficial packages can be downloaded here), and you can start the software by using a command line passing the .jar binary to javaw.exe. Shortcuts and bat files can of course automate this. This wouldn't even require a "real" Java installation.

For a website with a Java applet, I'm not entirely sure, but it would need a proper installation of the required version. I would take care with this path, as the Java plugin for browsers is the precise place where some security vulnerabilities appear from time to time. I would always try the latest version, only going to a former one if it doesn't works.