Upvote:-1
You do not need a VISA to travel to Europe. Your passport will be checked twice. https://travel.state.gov/content/travel/en/international-travel/before-you-go/schengen.html
Unfortunately Croatia is not yet part of the Schengen Agreement.
Onwards, when you arrive Vienna and when you leave Vienna. That 1:25 is a bit short, but under normal circumstances it should be enough. If you're travelling with one ticket, you should ask your airline / agency if they provide support or what kind of rights you have with that ticket.
In Zagreb, there is no passport control for internal flights, but you should have your passport with you all the time, it is also needed when you show your boarding card.
Flying home, your passport will be checked first in Dubrovnik, than in Vienna and depending on your ticket and airways company maybe in Munich. If you have a trasfer ticket, probably won't.
Upvote:0
Going from Chicago to Dubrovnik, the only border check will be entering Croatia in Zagreb.
Going from Dubrovnik to Chicago, you will clear a Croatian exit check in Dubrovnik, a Schengen entry check in Vienna, a Schengen exit check in Munich, and finally the US entry check in Chicago.
You will not need any visas, and should you miss any connection due to delays you'll be re-booked onto the next one, assuming the whole trip is in a single booking.
Upvote:4
On your outbound trip you will probablly not pass through passport control at Vienna since both your incoming and outgoing flights are outside Schengen.
In Zagreb you will be entering Croatia, so you will need to pass through their controls. Your flight from Zagreb to Dubrovnik will then be a domestic flight.
On your return trip you will be leaving Croatia in Dubrovnik. I'm not sure if Croatia does exit checks.
You will then enter the Schengen area in Vienna to catch a Schengen internal flight. So you will have to go through Schengen entry checks and will leave it in Munich so you will have to go through Schengen exit checks.
And finally when you arrive back in the USA you will meet US border control.
In addition to government run immigration controls it is likely that at least some of the airlines will check your passport before letting you board the flight.
As a US citizen under normal circumstances you will not need a visa for any of this.