Couple of things you should take into account, the doors are quite long, front to back, so the likelihood of bass vibration is very real.
Find some stiff 2" diameter cardboard tubing, cut it down to fit right at the point of where the woofer installs, make sure it will not interfere with any moving parts; window glass, and lifting mechanisms.
If you find a spot or two close to this area that will not interfere, use epoxy glue on both sides of the tube (s) and reinforce the inner and outer panels in this manner.
Do a bit of test fitting before moving on to gluing the tubing in place, and once glued, allow ample time for the epoxy to fully cure.
Enclosing the sub with any material to create a "box effect" will increase resonance (the aspect of deep bass vibration) Woofers, by design, move a large amount of "air" enclosing is only going to add deeper resonance...
If you have built some sort of enclosure surrounding the woofers, make SURE you have also allowed an escape route for this air mass through a sound tube ( a tube that allows this enclosed air to be pushed out freely as the woofer operates.
Most home quality speaker enclosures come with sound tubes installed in the back, if you place your hand at the back of these sound tubes while the speaker is operating, you will notice a HUGE amount of "air" being pushed out...
You also have to take a closer look at the interior door design, and see what is causing this unwanted bass vibration.
One way to do this is to create a CD recording of just bass at a sustained note, then continue recording at lower/higher frequencies... forget about mids and highs, those are frequencies that usually do not cause vibrations...then you can load this CD and play it with the woofer hooked up and feel/look where the vibration is being caused.
For comparison, I left my door woofers factory speakers, ( shaker 500) did not upgrade them, and used Dynamat inside the door panels, I upgraded all 5X7's throughout the car, front and back...got an amp in the wheel well powering a 10" Cerwin Vega woofer in a removable enclosure, with dual air mass tubes built into the enclosure for bass escapement, ZERO distortion OR vibration.
My entire upgraded sound system was done by a close friend of mine that is the head installer for Best Buy here in Toronto... sounds far superior to the stock system that came with the car.
Hope some of these suggestions help you somewhat.
2011 GTCS Candy Red;Premium leather interior package, Remote start, security package, upgraded speakers, 6MT 3:55. 35% tint all around, Borla "S" axle backs, & clear corners.
The rest is all FORD