Is there any point at which the distance becomes too large to the extreme where you basically get “deleted” from existence?

This is basically what the definition of “observable universe” is. It is the part of the universe that is close enough in space and time for light to reach us. So if you say they get transported to the observable part of the universe, then yes, their signals will eventually reach earth. But the closer they are to the edge of the observable universe, the longer the signals will take to reach, and the more red shifted they will be due to the expansion of the intermediate space as the signals travel to Earth.

Note that there are some semantics at play; “observable universe” might refer to the parts of the universe that have emitted light in the past that is reaching earth now. But the the light emitted by those places now might never reach Earth because they are now too far away. So if these astronauts got sent to one of those places then no, their signals would not reach earth.