In the first sentence are you trying to say there is a team member position available? If yes - then say “There is a vacant team member position available.”
If you are referring to a vacancy left by a current team member then state: “One of our team members left, leaving a vacant position.”
The second sentence is correct.
The third should read: We did not have internet in the past, so we hand to rush home in time to watch our favorite show.”
The forth is confusing out of context. I would state it - We can communicate faster on the internet.