I thought "surely AndroidTV must be able to do this", but it appears not to be so (relevant discussion thread), even though the NVidia Shield comes with a camera. They do seem to support video chat over Twitch, but not others. Weird.

One other option is a Chromebox. These came out in 2014 and don't appear to have been updated since then, which is always a red flag, but $140-160 gets you a PC-in-a-box which is relatively hard for somebody to screw up, since all it does it run a web browser. Any video chat that uses HTML5 goodies will work with it.

At least a year ago, Google was pitching this explicitly as a low-cost video chat solution for business.

A cheaper solution is to buy a Chromecast device ($35-ish) and cast to it from a laptop running your video chat. This will definitely work for Hangouts running in Chrome (instructions here). You'd need some kind of third-party screen-scraping-to-Chromecast solution to make it work for Skype.