People can die of stress. Not sure about a “sickness.”

I’ve had friends who have thrown up while crying hard. I don’t think they get an illness but they say it made them sick.

You’d have to cry for a terribly long time in order to dehydrate a body. I can’t imagine crying for that long!!!

Yes
https://www.healthline.com/health/i-cant-stop-crying
Mood disorder may be causing this and treatment should be sought to test for other reasons.
There is a list on the link of possible causes and to get proper treatment to alleviate the distress that it causes.

I just threw up from crying after my second mental breakdown today. Stress can make you ill. Crying involves stress.

^ I throw up almost daily, from stress. Not usually crying involved in the vomiting, but it is triggered by stress/anxiety.