Justin,
This was a common problem and it even has its own TSB. It is TSB #04-10-2, Body - Rear License Plate Loose. It can be repaired under warranty. However, if warranty is up then you can fix it yourself. Here is the TSB in full:
ISSUE
Some 2000-2002 Lincoln LS vehicles may exhibit loose rear license plate fasteners. This may be due to the attachment area of the license plate housing breaking (the holes where the license plate expansion nuts mount Figure 1).
ACTION
To service, enlarge the license plate expansion nut attachment areas and install new expansion nuts. Refer to the following Service Procedure.
SERVICE PROCEDURE
NOTE REPLACEMENT OR REMOVAL OF THE REAR LICENSE PLATE HOUSING FROM THE VEHICLE IS NOT REQUIRED FOR THIS REPAIR.
1. Remove the rear license plate from vehicle.
2. Remove the two (2) expansion nuts (license plate attachment nuts) from the rear license plate housing (Figure 1).
3. Enlarge the two (2) rear license plate expansion nut holes up to 17 mm using a file or suitable tool (Figure 2).
4. Install two (2) new rear license plate expansion nuts (W710640-S300) into the enlarged holes. The new expansion nuts should be installed with the retaining tabs in the horizontal position for best results.
NOTE DO NOT USE A HAMMER OR OTHER SUCH TOOL TO POUND THE NEW EXPANSION NUTS INTO POSITION AS THIS WILL RESULT IN DAMAGE TO THE HOUSING. THE NEW EXPANSION NUTS SHOULD BE SNAPPED INTO POSITION BY HAND ONLY. HOLES IN THE HOUSING MAY NOT BE ENLARGED ENOUGH IF EXPANSION NUTS DO NOT SNAP IN BY HAND. IF NECESSARY ENLARGE THE HOLES SLIGHTLY BUT DO NOT EXCEED 17 MM.
5. Reattach rear license plate using two (2) screws Part Number N807393-S61.