This is a difficult problem, and I fear that there is not going to be a solution that satisfies your "inexpensive" requirement. I can tell you that IF there is a cheap solution, and I doubt that there is, then that solution would likely be a mechanical solution and not an electrical solution. Since this is an electrical web site, that's beyond our scope. Given that, here is my solution with a heavy dose of electronics...
Your requirements are: A sensor on every wheel to detect position and something to start/stop the wheel from spinning. A requirement that you didn't mention is that the overall complexity needs to be kept simple. A simple system will be much more reliable than a complex system. Also, mechanical wear and tear will be important if you want this thing to last a while.
So... I immediately throw out any idea of using motors, gears, or solenoids. The mechanical design required to hook this all up will be complex and prone to failure.
Instead, make each wheel into a motor. Start by placing the wheel on a smooth shaft. Use ball-bearings on the wheels so they run smoothly. The shaft itself does not rotate. The wheel needs to be made of something non-conductive and non-magnetic, like plastic.
The wheel has mounted on it 4 or 8 permanent magnets. I'm not sure if it is 4 or 8, as I didn't go through the design to that level of detail. I'm guessing 4 if you can align them so the North pole faces the South pole of the next magnet, and the poles are evenly spaced around the wheel.
Mounted near the wheel, on a bar that goes across all wheels, are more permanent magnets. One magnet per wheel. This magnet shouldn't be too strong. The main purpose of this magnet is to force the wheels, when stopped, to align properly so each symbol is perfectly lined up with the next. Also, if the wheels are not perfectly balanced then this magnet keeps the wheel from rotating so the heaviest end is always down.
Also mounted near the wheel, mounted on a PCB, are two electro-magnets per wheel. These form the coils of our motor. Drive electronics, probably an H-Bridge, are also mounted on the PCB.
An IR LED and Photodiode pair, also on a PCB and one per wheel, is used to detect the position of the wheel. Somewhere there is a single black mark on the wheel, and the sensor picks that up to identify "symbol #1".
Each PCB will also have a single microcontroller to drive the "motors" and sense the position. The Microcontroller code is challenging to write, but not impossible. Due to the size, I would guess that each PCB only supports 10 wheels. So for 100 wheels you would need 10 PCB's. This means that some form of communication between PCB's is required, but that is relatively easy.
The electro-magnets will probably need to be hand built, but everything else is just a PCB.
The whole unit would be fairly easy to assemble, and since only the ball-bearings are "moving", everything should be quite reliable.