I see a lot of mis-information here. I custom rebuild these units and can tell you I have had to replace many gear sets from over adjustment procedures. Following some of those listed will cause you problems.
Here are some facts:
As John Z mentioned the vette boxes are unique to corvette and the box is the same for both PS and manual cars.
There is an on the car adjustment procedure but I don't recommend it.The best way to adjust it is with the box out. chances are you're going to need more then an adjustment as well. Some of these boxes are sloppy. Attempting to adjust it without a 0-30 in/lb dial TW is a waste of time and money. Fish scales and clickers are not the correct tool to use.
Manual steering cars get beat up more then PS so chances are someone already "adjusted" it. Look at the lash screw in the top and see if there are any threads showing above the nut. If not then your gears are done.
MANY of the rebuilt boxes on the market are cheap rebuilds and not worth the $130-$180 charged. I have repaired many of these already. Most consist of a paint job and over greasing. I rarely find the internal parts replaced and there is no way new gears are in any of those jobs.
In my updated thread on boxes I show where one rebuilder welded the gears and another installed the worm nut upside down. Not the type of work I would recommend.
I suspect the recent modification of installing aftermarket or other make boxes is a result of the current lousy work on stock boxes. In fact a properly rebuilt box,blueprinted beyond the typical specs preforms 100% better then a stock box and 1000X better then a common rebuild. In addition they fit the car without modifications and hold a higher value if the car is sold. I'd put a BP box up against any others.
Now as mentioned and it was good advice, check the suspension and rag joint. The parts should be tight. Center the box with the car in the air and have someone move the wheel while you look at the rag and box input. There should be no lost motion with the box on-center. Off center- yes. Rag should not be "S" shaped,ripped, or dry rotted. Rag rebuild kits on the market are cheap junk, use only a new GM rag or if you have a 63-66 have that one rebuilt by the place that does them for the vendors. You can PM for that link so we don't PO anyone.
The large lock ring secures the nut which sets the preload. The bearing preload should be 3-5 in/lbs. With new gears I set them at 5, with rebuilt worm or original they will be lower. Once the preload is set then set the lash. If you adjust the lash off center you will wreck the gear in time.
I suggest you look up my thread and link on boxes before you attempt any adjustments, but it is your car so good luck. If you need help let me know. The boxes I build are the best anywhere. Many are surprised just how good a properly setup box,suspensions and alignment can make a vette handle. I can't tell you how many times I've heard old cars just didn't handle or brake well. Those same people never drove a good vette I guess.