Exterior adhesive

My 1998 Camry has a problem where the window won’t fully close on the back edge of the drivers window. (Picture below, the second picture is the passenger side window)This makes it impossible to use car washes and to fix it I was thinking of just taking a piece of cast acrylic and attach it to the door just behind the window and cover the small gap that’s open. But I’m not sure what kind of adhesive to use to make it stick long term and under the conditions of driving

I don’t think the addition of a piece of acrylic will work very well. The gap is due to the window regulator being worn, glass binding, or in need of adjustment. The door panel needs to comes off.
My suggestion would be to dig in or have it dug into and fix it right.

If you absolutely, positively do not want to go into the door, a small piece of foam tape might keep out the water and would work better than gluing a piece of plastic in there.

The drivers door glass is on top of the glass run (weather strip) rather than inside of it. If the door glass tilts out of position due to some failing part the glass may pop out and ride outside of the glass run.

It may be possible to adjust the glass position and straighten the glass run. Worst case it may be necessary to replace both the glass run and window regulator. I have been able to manually guide the glass into position for customers that were not interested in repairing it, the window should not be lowered again until repaired.

I agree with taking off the door panel and trying to fix this correctly. I think it’ll be hard to rig something that can hold up against the pressure of a car wash.

The inside panel needs to be removed and the window properly readjusted or repaired.
The dealer’s parts guy should be willing to print a procedure for removing the inner panel if you ask nicely. It makes removing the panel far easier if you know how it’s installed.