Central Vac

That is going to be tough. It is either clogged or not connected. Since it turns on it appears it was intended to be connected.

Can you hear what sounds like the vac when you turn it on coming through the pipe?

Try fishing a wire or something down the pipe. The sharpest bend should be right at the outlet. How far can you get?

My guess is you will find a kid's ball or some such thing in it somewhere.

Sounds to me like they forgot to connect it at one of the T's or elbows. Hopefully it won't be in the wall somewhere....

Try running a snake into it. You could use a garden hose.

We had a central vac in our previous home.

When things "went wrong" and a tube got plugged, I sometimes used a hose from a portable vac to apply sucktion to the non-working outlet.

You can go further (I didn't have to, BTW): if you have an air compressor you can apply pressure at the central vac end of the tubing and have your kids (or whoever) hold down the flaps on the working outlet(s) and let the preesure work on the clog.

Using either technique, I suspect, will likely reveal that the "rough in" was sloppy and the outlet may not actually be connected in the first place.

The motor part of a central vac system can be had for something on the order of $100 (cheap unit). I don't understand why someone would install the tubes but not the central unit. There is something fishy.

Not so fishy. When I had my house built, money was tight and there was no extra to spare. I had the roughed in vacuum plumbing put in before the walls were closed in. After a year, I had enough equity in the house to remortgage and buy the rest of the vacuum system.