Usually these systems are integrated by the software companies themselves. So, WebELS would write a Moodle Plugin to handle the integration and submit it to Moodle.Org for inclusion in their plugins database.
Companies often do this in the hope that they will get more Moodle users interested in their product.
One example of this which looks very similar to WebELS is Big Blue Button:
That is free open source software and they have written Moodle plugins already:
You can get affordable hosting from one of their partners:
So, my approach would be to ask WebELS to write a plugin, and if they wont consider switching to BigBlueButton or some other supported platform.
I will say recording of sessions is not great in BBB, but the platform is being developed and improving.
On a side note Adobe Connect also have Moodle plugins but that is a much more expensive service.