Wouldn't see why you would even want to use a computer interface, it would just be more difficult to get it up and running. For one, those motors would have to be driven by relays. You would then need to buy a relay board for the PC, then write custom software for it. If you are proficient at electronics and can program decently, you could write the program for it and then build the circuit. This is similar to what I did with my skeeball. I would suggest just designing how it works first, then figure what gears/motors you need, etc.
For instance,most of those once you go down to a certain point it continues on down, grabs whatever (always air in my case), then returns to the top. Something could be mounted on the chain to trip a momentary throw switch that goes to a flip-flop, which cuts off the players connection and grabs the item. When it gets to the end, it hits something to reverse voltage causing it to return.
It looks like it has a clock, so you could use some type of electronic timer for that. After a certain time, its cuts the players power and/or does the grab.
If you are wanting to do sound effects and whatnot through a computer, then that is another issue. I could have done my whole skeeball with just circuits, but since I want to have sound effects, I just interfaced it to the machine through RandyT's Keywiz, using the same buttons he is using for my program as well (that way I can test both). What the program does is use the data lines on the printer port to control the scoreboard and ball count. A button on the front of my machine resets the ball count/score. The program scans the keyboard, waiting for certain keystrokes. When those are hit (via the ball scoreing rings), it increases the ball count, then slowly increases the score (so you see it add - at full PC speed you don't see it). It then plays a sound depending on the ring, and if you get certain scores it does other sounds as well.
The big difference in mine and RandyT's program is that Randy's utilizes a monitor, mine uses the custom circuit. I have some other chips coming in that will probably work better, as those will let you drive several 7 segment displays and have the PC tell it what to display via serial connection. That would make the whole display be only two chips (plus misc parts) instead of 7 which is what mine is now.
Just plan, plan, and plan. Nothing more important than that when your the first one to attempt to DIY something like this. I've probably wasted about $50 in misc things as I went along - parts not needed, circuit redesigns, or the most recent was forgetting that one chunk of my LED's didn't have the resistor on the cathode, so I smoked them when I turned them on. I tried some cheapies at Radio Shack, but they were not even close to the brightness or quality of the ones I had. You can barely see them lit. So, I get to order another pack of them for about half the price of Radio Shack. I will end up getting a relay and probably a solinoid as well, and try to automate the entire machine since I am far enough along in it.
Almost forgot... this is supposed to be about your project
My bad. I got carried away I guess...
Look up things online about parallel port interfaces, that should help you if you are going to try the PC route.
Brad