No, not quite.
This is the one of FIRST Choice I was talking about. The item you linked if for getting an ethernet port on a computer that doesn't have one. What you need is something that goes on the robot so that more than two devices can be plugged into the robot at a time. The one above is light, cheap (can't beat just shipping), easy to use, and pretty small. You'll just need a DC power plug (like the one for the radio, might even be the same size) and plug it into the VRM.
Despite the fact that you only have two devices, and the built in radio has two ports, I still highly recommend you use a switch. In the pits at competition, you'll have to use either USB or Ethernet to communicate with your robot, and USB has issues at longer distances, such as testing on the practice field (most events don't provide a practice radio like FIRST mentions). You'll really want at least one free ethernet port to do debugging on.