Recently bought a 2003 Grand Caravan Sport (3.8L) with 137k on it. First thing I did was put the two new rear tires on the front, which required installing 3 new wheel studs, thanks to the tire shop's failure to hand start the lugs before using their impact - GRR.
Front end had all kinds of shakes and rattles over bumps. Replaced sway bar bushings and sway bar links, which tightened up the front end, but I then had a vibration - almost a wobble - at various speeds and usually on flat pavement which wasn't coming through the steering wheel. Mechanic friend said strut bushings can cause this, and the struts were original, so I replaced them with Monroe struts and aftermarket bushings. Ran the strut rod nut down tight with an air impact after installation, which required pulling the wiper box. Nice design changes in the wiper box over my old Gen 3 T&C. Anyway...
I still have the vibration, and at highway speeds or any speed above about 30mph, so it doesn't seem to be transmission related. Moving the steering wheel left and right with the engine off creates a slight clicking feel (not sound) when a hand is placed on top of either front tire. There is very slight left/right play in the tires when one is off the ground, so my next stop was tie rod ends.
Outer tie rod ends didn't move when the tires were manually moved left/right, but there was a little movement in the inner tie rod bellows. Pulled the outer tie rods out of the steering knuckle, and could get zero play from pushing/pulling the inner tie rod ends on either side. Reinstalled the outers in the steering knuckles, and with the front on jack stands, checked for independent play by moving one tire left/right and watching the other tire. They moved together, with no visible play.
Having second thoughts on my diagnosis, I did NOT install inner tie rod ends. If the steering rack needs replacement, they come with inners installed, but I'd expect a bad rack to leak fluid, and this one doesn't. I don't feel the vibration in the steering wheel, but in the floorboard, so I started looking elsewhere. Checked the new sway bar links and bushings for looseness, but they're tight. Checked the half-shaft to steering knuckle nuts for a possible worn bearing, but they were properly torqued and there's no in/out play in the tires when mounted, just left/right, so that's out. Odd that I wouldn't feel a tie rod issue in the steering.
Thought about possible worn rack mounting bushings, but there are none on the GC that I can find.
Could I have a bad steering rack, a junk motor mount, or worn out rear sway bar links? Replaced the rear shocks with the front struts, and I can't find two aftermarket rear links locally, or I'd have done them already. I haven't put the worn rear tires back on the front, since a vibration caused by tire balance or bad belt would likely get worse with speed, and this one doesn't.
I'm about out of patience, and about out of money too (tax time). It's probably obvious to someone here; what am I missing?

 

UPDATE:
Not replacing inner tie rod ends was a good call. After taking the boots off the steering rack and looking at the inners, it became obvious that they were tight, and the rack itself was sloppy.
Replaced steering rack, no movement in either front tire with the other on the ground now.
Alignment went flawlessly, but the vibration remained.
Took it to a Chrysler mechanic friend who took it for a test drive, and he immediately noticed the vibration occurring when the torque converter started to lock up. This happened in both 3rd and 4th gear, at about 35 and 45 mph. He called it "torque converter lock-up shudder" and said it's common. Chrysler shops usually R&R the converter, which involves pulling the transmission; not something I'm equipped to do in my driveway with limited clearances and without a lift.
He suggested doing a trans filter and fluid flush; that might help if the converter isn't too worn.
I'll keep you advised.

 

Thanks for posting the information about what you've done so far. Always helpful to read these things from other DIYers.

Hopefully your mechanic friend didn't recommend doing a power flush on the tranny - it's often the death kiss for these vans. Some guys do a fluid transfer using the van's pump via the cooler hoses, which would be fine. But most of us just drop the pan and refill the 4+ quarts of fluid, along with the filter change. And repeating the drop fairly soon will further "improve" the quality of the fluid. Some guys also install a drain plug or new pan with a built-in plug to make subsequent fluid swaps easier.
There should be only a thin layer of sludge on the bottom of the pan and magnet - anything else would be bad news. And the fluid for these 4 gens is ATF+4 - no other type or additive.

 

Just to add to cv's post, instead of dropping the pan a second time just pump the fluid out the dip stick with a $20 - $30 pump available at most parts stores.