Vaporetto pass

Yes, you can buy it at a desk in the airport or from a kiosk outside by the bus pick up point.

Make sure you validate your pass each and every time before boarding

Fines are high and they do check

I was in Venice last week and saw multiple ticket inspectors over the course of several days.

If you are taking a bus from the airport, there is a big vaparetto ticket office at the entrance of Venice. The buses all come here. Piazzale Roma vaparetto stop. The train station is just across the bridge. We found this to be the easiest place to buy a ticket. Staffed with real people who were very helpful. We were staying near here which made it more convenient. We could get anywhere from here, too.

Of course, if you are taking a bus, you can get a discount on the bus ticket as part of the vaporetto pass. In that case, you will definitely want to buy at the airport.

I'll be in Venice in a couple weeks and my family (2 adults, 2 kids) will be purchasing a 3 day pass for everyone. What's the easiest way to manage all four passes? Since we'll be travelling altogether, I thought it would be best to have it on one person's phone app. Or we could print out all 4 and have one person carry them all.

Recommendations?

At Marco Polo, vending machines.
Buying Tickets on the Spot
if you haven't bought a pass in advance, the machines allow you to purchase:
Single-use urban transit tickets
Multi-day Vaporetto (water bus) passes (1, 2, 3, or 7 days)
The Aerobus airport supplement (Line 5 to/from Piazzale Roma)

Replying to Nixit.
I think that hard copy of your passes will be easier than using a phone.

Scroll down to the airport arrivals layout showing location of the Venezia Unica desk where you can buy passes with the bus adder. As you exit the baggage claim, also location of kiosks (either before you exit baggage claim or outside the exit. It shows the bus stop location, you want the No 5 Aerobus.
https://www.veneziaunica.it/en/plan-your-trip/getting-to-venice/marco-polo-airport

Nixit71 --- our grandchildren aged 9 and 6 absolutely loved having their own vaporetto passes to beep at the readers (our son kept the passes with him between vaporetto rides).