There is really only one official source for footage of FRC competitions. That is the FIRST YouTube channel. The channel is updated with new videos every week and playlists are made up for each event. The playlists are organized by match number and are usually updated within a day or two of the event. From what I have seen, the public archive goes back to 2016 for championships and 2017 they began to record all events.
Prior to 2016, the only way to get footage of a competition was to have someone record it for you. This was usually done by a parent or mentor. The quality of the footage was usually not great and it was not always easy to find. At one point there was a website called FIRST Video Archive that is available via the wayback machine still, but otherwise gone. This YouTube channel FRC Video Archive has some of the videos from there! This is helping keep some of the history of FRC alive.
The Blue Alliance has a live stream program called Gameday. You can chat, watch multiple events at once and more!