Can you change the crankshaft in a 2003 dodge neon without removing the engine from the car?

Yes, but it would take up to four times longer for an experienced mechanic. To do this properly you would need to remove the engine. If you have a manual, you would also have to separate the engine from the transmission so you can separated the clutch from the flywheel. If you truly need a crank, the "crank" doesn't normally go out on it's own. There are almost always other issues that should be checked...rings, bearings, etc. It is also unusual for such a young (2003 model) engine to have crank issues.

The easiest way to perform what you're asking to do would be on a car hoist. You would need to take almost every step that removing the engine would require, but instead of removing the engine, you could "drop" the engine cradle, sometimes called the K-member, while supporting the engine. You could then work on what's called the "bottom end" of the engine.

Lastly,It is safer and easier to work on the engine when it is out of the bay.