Budget accommodations for families are often possible outside major cities. Rather than locate two separate places for 5 that are likely outside your budget, could you instead take a single base town in between, then travel into these cities for sightseeing? (That's what I'd do kids on my normal every-man's travel budget if I had 3 kids.)
Prien is right in between Munich and Salzburg, on Herrenchiemsee Lake (which offers both recreation and Bavarian King Ludwig II's Herrenchiemsee Palace) and also on the main train line. From a Prien base, your entire family could take a direct train into the city on day trips for €31/day using the Bayern Ticket.
Prien lists 119 apartments at its local tourist office website. Enter your dates and your occupant information and see what's available. ( I came up with 4 places available for a family with 3 kids, using a set of arbitrary dates, July 11-18.) You will get prices and a few words in English - but the apartment descriptions will be in German and you will need to translate the text. You can also google the name or address of the apartment to see if there's a homepage for the place (and perhaps an English-language description.)
This place in Rimsting (a neighbor town to Prien) has one apartment that works for 5 and shows availability in July:
https://tportal.toubiz.de/prien/ukv/house/DEU00000060011066340
This brochure for Prien has over 70 pages with a complete set of listings but is in German.
You can look at other towns in the area as well. Perhaps Bernau or Aschau or a town elsewhere on the lake would work for you.
One commercial listing site with an English-language option that is user-friendly is www.vacation-apartments.com. The mother site (in German - traum-ferienwohnungen.de ) has a lot more listings but the English version may have what you need. Listings like the one below in Bernau have an availability calendar you can check. This place in Bernau accommodates 5 and shows availability throughout July.
https://www.vacation-apartments.com/33030/
Be sure about a place before you book. Generally, cancellation isn't possible unless the owner has a cancellation policy - be sure to ask about that. Some owners will request a deposit, but many do not, which makes booking pretty easy - but either way, you can be held legally responsible for the rental fee unless there's an agreement about cancellations.