Without actually seeing the cart, slots and items it’s hard to say what would be best. However my experience suggests that no single sensor type will work unless the cart is designed around it. e.g., a photosensor as darrelg suggests may work… right up until you insert an object that happens to be transparent to infrared. Likewise, microswitches seem like a good idea, but the slot needs to be designed around the switch. Neither of those will be under $1/point.
From a hobbyist perspective it’s easy to get misled by cost: wire, sensors, connectors etc. are cheap and time is free.
From a business perspective, integrating them is far from free and you’ll likely want more expensive parts because the cost of replacing them is high. For example, I make a product that has a cable harness with 50+ wires. Even in the quantities I need to buy parts, a single wire with contacts crimped on each end is about 75 cents. Then there’s the labor cost of just plugging that wire into a cable housing and testing it. Already over $1/pin for a simple cable. If you buy a simple cheap IR photointerrupter sensor from SparkFun, then you need to have bracketry custom made to hold it in place; you need cabling, connectors, wire dressing to route the cables. By that point, you may as well buy an industrial sensor that has all that done for you at a cost of $50. Haven’t even gotten to the labor of connecting everything up.
I’m not trying to throw a damper on anything, just being realistic about true costs.
IMHO, the only way you have a snowball’s chance of hitting a sub $10/slot target is with the most dead-simple approach: a toggle switch/pushbutton at each slot and the operator presses it when he puts something in a slot. That may hit the cost target but it will have to be wired. Emphasis on may. In reality, you’ll probably end up needing a pretty rugged switch and those won’t come cheap. It is possible to build a wireless solution for each slot, but unless you’re going into the business of building these things, you probably will never recover the cost.
Sorry, but while it may be possible to prototype this for $10 per slot, I doubt a warehouse floor ready solution will be that cheap. The electronics are pretty simple: an Arduino, BeagleBone, or RPi with I/O expander will read the inputs just fine. The problem is that something that has to be pushed many times a day, that will probably have things dropped on it, get dirty over time, etc. is typically not going to be cheap.