I know the DGT boards are used for this. But, do they all do so? Does every chess website connect their server to the DGT board and relay the moves? Or do they simply "steal" the moves from another site?
That depends on the tournament organisation. If the tournament organiser is happy, they provide an online PGN file which is updated everytime the player makes a move. A good digital board software can easily do that. Broadcast websites such as chess24.com simply has to poll that PGN file. Easily scaled to any number of chess boards.
Obviously, the last strategy requires cooperation. The 2018 world chess championship has no such facility. The broadcast websites has to hire someone constantly monitor the offical website (they have to pay for subscription), and relay the moves manually.
Generally, organisers especially open competition are more than happy to assist. It's a nice way for promoting their events. Chess moves don't make money.
They "steal" the moves. The organizers of the last world championship didn't like that and sued, but lost. See some discussion at What are the laws concerning copyright of chess games?. In short, the judge said that "it is well-established that sports scores and events, like players' moves in the Championship, are facts not protectable by copyright".
I can't speak for every site, but in my experience you can tell that they are doing it this way based on comments like "I just heard there was a move" and by the fact that the clocks are generally wrong. Also, why would the organizer make it easy for sites which they see as freeloaders?
(Of course, this depends on the event. I wrote the above thinking of high-profile events such as the world championship which is taking place as of this writing, and which I thought might have been on the questioner's mind.)