Astronauts use a communication network that NASA has setup and owns.
The way that astronauts communicate with earth is through the use of big radio antennas which are known as the Deep Space Network (DSN) which is basically a collection of big radio antennas.
Then they have communication equipment in the spacecraft and International Space Station that communicates from space through the antennas that reach earth.
Then in the NASA offices they have a station where they operate equipment and monitor the conditions in space and communicate with the astronauts.
That is how they also get the internet in space.
The internet in space goes through those radio antennas as well.