When flying from Venice to Paris and onto Chicago, do we need to go through customs at each stop or just when we arrive back in the United States?
'customs'--just when you arrive back in the United States.
There will be passport processing to exit the Schengen zone in Paris
Immigration and customs are two different things. (Americans have the bad habit of calling everything "customs.")
Immigration is for people, customs is for your stuff.
In Paris, you will go through exit immigration and get your passport stamped showing you are exiting Schengen.
In Chicago, you will go through both immigration and customs.
Americans have the bad habit of calling everything "customs."
Probably should be no surprise since the US agency is named "Customs and Border Protection", that it gets shortened to 'customs'
https://www.cbp.gov/