How do chess websites relay the moves for tournaments? 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.