Ah, I see. Yes in vSphere 6 and beyond it became mandatory to use SSL and we wend away from the java based client.
Are you still working in vSphere 5.5? If so, I'm assuming it's because of some type of security or approval restriction. That's unfortunate, but I've been there.
Yes, I meant the ESXi or VCS root account. But with this new info, you can ignore that line of thought for now.
So are they able to use the web interface? It also sounds like its an API call for this EMC backup interface. The web interface was OK in vSphere 5.x but much better from v6.x onward. If there was an upgrade to the java engine, it could have closed off an API port. Similarly, after a power outage, if there was a recovery of an old config file, or even replacing with a new config file (on the network/firewall side) that could also explain why the API call is cut off.
And also, based on your research, it's also VERY possible that the certificate for the java based vSphere client has finally expired and that is really hard to overcome. You'll want to weigh the impacts of not having a current back-up service running against upgrading vSphere (to at least v6.7 if not all the way to v8) or implementing a different back-up solution. Or, if you cannot upgrade vSphere (just know more problems will begin to surface as 5.5 is almost out of it's support window) perhaps there is a different way to scrip the backup from the EMC side?
I know, I'm kinda throwing out a lot of ideas, but not knowing your infrastructure in detail, I want to provide options based on the situation.