Feasibility of Amsterdam/Berlin/Munich/Zurich Trip

I assume you mean ONE Of those places not all in one week, right? ALL of them in one week would be very difficult and not very practical.

You could easily do a week in any one of those places (and add day trips or a second smaller town). It might be possible to do two of these cities together in one quick week. Up to you, what you want.

To save money, I'd not go in the summer which is peak tourist season in most of those places. (Switzerland is very expensive, and many would not suggest Zurich as the best place in Switzerland to visit, though I enjoyed a brief visit there.) Too late for this July, but you can take advantage of credit card sign-up bonuses to get miles or points to get award tickets to Europe.

Based on the sentence construction, my interpretation was OP is trying to do all of them.

I hope I am wrong

At this point airfare is not going to be bargain for travel in July and most of the mid-range priced accommodations are likely gone.

Planning farther ahead or going 'off season', or both are good money saving tips. Probably not much help for this trip.

What are some ways to save money on a trip to Europe as we are trying to stay on a $5000 budget

cut out Munich (it is way too far) and above all Zurich (and even more too far + one of the most expensive cities in Europe).

How many days exactly will you be there? Most here will advise staying in a given location for 3 nights. So, you might consider flying to Amsterdam, staying 3 nights, flying to Berlin, etc. But all depends on the actual length of stay.

I would recommend a week in Berlin. I had 6 nights there last year and didn’t get chance to see everything or take any day trips. Plenty of modern history, good transport and good value if on a budget - Amsterdam is expensive.

If you're on a budget, you aren't making it easy on yourselves. You're going in summer, when airfare and hotel prices are highest. And you've selected major cities, which are more expensive than smaller cities or towns. And you're adding $$$ for intra European travel because you're trying to see multiple (relatively) far flung places. And then, of course, there's the fact that you're trying to cram several weeks worth of places into a single week.

Pick one (or 2 relatively close) places. Amsterdam (and perhaps Bruges) would allow you to explore the Netherlands and Belgium. Or Berlin and Munich. Forget Zurich. There are other places in Switzerland that are better. And Switzerland is very expensive.

Rick has lots of tips for inexpensive sleeping and eating here: https://www.ricksteves.com/travel-tips/sleeping-eating. Low or moderately priced hotels are likely heavily booked up by now. You might want to look at hostels instead. Or look at renting a private room through Airbnb or VRBO.

"We identified Amsterdam, Berlin, Munich, and Zurich (and potentially other smaller cities in Germany.)"

That's a 2-week trip.

Maybe fly into A'dam, then fly to Berlin or Munich and fly home from there. That's a snug 1-week trip.

If you want to stay on the ground... and follow up on the idea of smaller places... Take the train from A'dam to the Middle Rhine Valley (stopping for several hours in the German city of Cologne.) Stay 3-4 nights in each area, then fly home out of Frankfurt.

In the Middle Rhine Valley, the best combination of scenery and interesting towns and castles lies south of Koblenz and north of Bingen. See map below. St. Goar is an especially good base town for train users, but Boppard, Oberwesel and Bacharach are also good. You can get to FRA airport from any of these towns by train:

https://www.mittelrheinbahn.de/ausflugsziele

(The black line is the railway; the dark blue line and the red line - the Rhine Castle trail - are scenic hiking trails. )

You will find both the Rhine-town accommodations and the transportation for this option to be relatively inexpensive. Advance-purchase train tickets from A'dam to St. Goar are selling for €25 - €30 one way on the random July date I just checked; use the "stopover" feature at the DB itinerary page to schedule in time for a stop in Cologne:

https://reiseauskunft.bahn.de//bin/query.exe/en

I wish that OP would check in and tell us what was decided.