I have a Sony VCT-14U tripod plate (the cheaper one Sony makes) and it is becoming very difficult for me to mount my camera into the tripod plate. I have to hold the tripod with my left hand, brace the back of the camera with my right hand and shove as hard as possible. This is the only way to get the locking mechanism to engage and lock the camera onto the tripod. Any ideas on what lubrication to use to lube the lock mechanism and how to go about lubricating the plate? Thanks in advance.

Hmmm, cheaper than what. The Sony and JVC version are the same plate. I've had and still have one that has been used for a loooong time with both Sony and JVC which is what currently rests on it and while once in a while I have to give the camera a good push (means I missed the mark and it's not 100% lined up) and I NEVER lube it.
Instead of thinking about using lube on the QR adjust the rear screw a bit looser (higher) which will give the rear U shaped plate on the camera a chance to grab the lock down. Not to loose as then the camera will wobble on the plate but maybe a half turn to start.
I really wouldn't use a lubricant on the plate for the purpose of making it easier to load the camera on it but that's just me.

If you look underneath the plate, there are 6 allen screws towards the front. Take these off and look inside. Chances are there is a ton of crud and/or rust. If not, check to see that the lever arm is not bent and binding somewhere. Clean out the inside, spray with WD 40 and you'll be good to go!

Thanks, Warren. I took the plate apart and shot the linkage up with WD-40. Seems to have worked.