If you are using the vehicle for only a short time (usually less than two weeks), the Parking and Transportation Office can issue you a temporary permit at no cost. However, if this is a long term change, then you will need to have a permit on it.
If you still have your permit from the original vehicle, you can switch the permit from the old vehicle to the new vehicle. The permits are made to be transferable between vehicles. You will have to update your account by logging in to The Permit Store website and clicking on ACCOUNT in the upper right hand corner. Add the new vehicle under the MANAGE VEHICLE link, then change the alignment of permit from the old vehicle to the new vehicle under VIEW PERMITS.
If you do not have your old permit or it tears apart when you attempt to move it, then you will need to purchase a Replacement Permit from the Parking and Transportation Office for $10. This option is only available through the office and is not online.