If I replaced the front wheel on my bike with a ball, would that work?

It seems you're thinking of having a big ball bearing at the front of the bike, rather than something that only rotates in one axis. So, no, it wouldn't work and the reasons should be obvious to you.

It works for a vacuum cleaner. I have one of the Dyson ones and it’s way better than the wheeled one.

There used to be a ball wheel barrow but I’ve not seen it for years. I suspect for the same reason , it wouldn’t be a good idea for a bike, it can move sideways. With a loaded barrow on a slope, this would be bad.

It would move like a supermarket trolley. Would you want to be riding that?

A ball has a lot of surface area and it creates friction with the road, so you have to supply enough power to overcome the friction before the bike can move. Bicycles come with skinny wheels and tires to minimize friction. A bike with fat tires and wheels is slower than a bike with skinny tires. That is why bikes in the bicycle races and road bikes have skinny tires. Fat tires give more comfortable rides because they can insulate the rider against shock when the tire hit a pot hole and other irregularities but they make the bike go slower  Road bike tires also recommend high air pressure, as much as 100 psi, and the high pressure allows the tire to minimize the amount of surface area that contacts the road, and minimize friction, but again it means the ride is less comfortable. The same principle applies to cars. You see skinny tires on econoboxes and the minimize friction, but you see fat tires on sports cars because fat tires allow the car to stay on the road better when going around corners at high speed. Since a sports car has a bigger engine than an econobox, it can supply enough power to overcome the friction created by fat tires. 

NOT WISE. but possible.

You can always use a smaller wheel. It changes the geometry and will feel weird. (Smaller bicycle wheel, a solid wheel, skateboard/roller skates/scooter wheels....) It just needs to be 4+ inches

No, that wouldn't work.

A bicycle front wheel has to be able to resist sideways forces. Otherwise, it’d be like a front wheel skid. Generally a very unpleasant experience for any rider of a 2-wheeled vehicle.

If the front ball is free to turn in any direction, how do you expect to be able to steer?A ball-shaped wheel on an axle would give better ”float” on soft surfaces. Wouldn’t do you much good if the rear would sink in past rideable depth. And there already are Fat Bikes to deal with soft surfaces.

All I can say is...  

In theory it might work though it wouldn't function as well as a wheel.  With a wheel it's axle can be much smaller than the wheel's diameter. This wouldn't be the case with a ball.

With an axle the ball can't just move in any direction. It could move in any direction if the forks where designed like a ball point pin. Ink also functions as a lubricant in a ball point pin. This means a lubricant would be necessary for it to function. Not very practical for something comes in contact with surfaces that have particle debris on them. The lubricant could also produce a hydroplaning effect as well.