Yes, there is support for recording video and surely you can send this data to the server.
There are many tutorials available, you can find them with a simple google search:
- https://influxis.com/simple-as3-recorder/
- https://code.google.com/p/flvrecorder/ --- has a sample
- http://www.purplesquirrels.com.au/2012/12/record-and-play-back-video-with-air-for-ios-on-ipad/
and many, many others.
If you want to use free server, where to send video to, I suggest to look into Red5, it seems that it's not supported for a while now, but the latest releases are running well.
Red5: http://www.red5.org