For a previous event, we were hosting livestreamed presentations on Zoom.
We used a vendor called Archive Captioning for real-time remote captions.

There are a couple of options for integration. One is referred to as remote
CART, and viewers who want to view the captions can view them at a distinct
URL. Zoom also provides a both a captioner "role" in Zoom that allows a
captioner to enter the data, as well as a captioning API for vendors who
are able to integrate with this system.

I'm less familiar with Gotomeeting's capabilities, but my understanding is
that they're pretty similar to Zoom's.

cheers,
Mark