To start a Rally, you need to be in a Crew. The Roomie you want to send out against the Rival should be at home.

From there, you can tap the Rival you want to start a Rally against, select the Roomie you would like to be the Rally leader, and the amount of time before the Rally should start.

When that time elapses, the Rally begins, with the Roomie and any other Roomies that Crew members have sent to support travelling to the Rival/RadChad's location to attempt to defeat him.

Good luck!