We have the same problem, but its more than one app that requires older Java to function. Most are supported by "big IT companies" that somehow can't work with up-to-date Java, so I feel your pain!
We let our affected users stay with Java 1.6x, but we also track those users, and reach out to them whenever there is a new Java update to see if they still need it. We also have other apps that help mitigate the risk, so we can "accept" the risk for now...and if it came to it, we could just block the software.
I did try to get management to opt for an RDS/VDI option to support legacy apps that need unsupported software like old Java, but it was just too expensive for us to take on.