Can you make yourself sick just by willing it?

Not exactly.

But, if you really don't want to do something, you can convince yourself that you "feel bad" and that you are "too tired" and "too achy" and "have a tummy ache" or whatever to the point that you almost feel like you have those things even though you was fine before you started thinking about it.

You wouldn't actually have a fever.  You probably wouldn't actually throw up or have diarrhea - but your body would feel miserable because you convinced yourself that you are "too sick" to go do whatever it was that you was suppose to do.

Not really but you can induce vomiting by putting your finger down your throat.

Not exactly,  but your brain certainly can and does.   You are aware of this phenomenon known as withdrawal.