Thanks, Onions.
There are some improvements I know I do not have time for but for future work, if anyone wants to do a project similar to mine, I would recommend buying motor couplers and attach the Knex rods to the motor shaft in that manner. They both have the same diameter but I felt the motor shaft was not long enough for what I needed and a coupler would give more power to the shaft. I will say this, though. The NEMA 23 bi-polar stepper motors I used fit almost perfectly with the Knex housing that I made for it which was a surprise.
Another would be if I had a larger budget, I would definitely have made this larger and order all-threads (not sure if I would go with plastic or metal) to move the motor along the axis rather than using the wheel/pulleys. I would make the claw machine larger in the sense that there would actually be a reward drop zone that people can only grab their prize from and they cannot reach up into it just like the real thing.
As for electrical hardware, I wish I had bought L298 Dual Full-Bridge Driver. I was having power issues with having all four motors "on" at once so I had to turn drivers off at certain times when not in use. Using a L298 would eliminate that along with providing an additional 24V DC power source.