(Continued) Vacation rentals are a great way to go. Book early. The town tourist office websites are usually your most complete resource, and the town TI will often give advice and help you book.
I'd start here to get an idea of what's possible: vacation rental website
Do a keyword search with "Germany", then "select all", and "6 persons", and 500 places that can accommodate your family will appear.
Town websites: these have a rentals page or a link to a regional tourist office that has rental listings. For a given town, try www.(town name).de then look for an English option. Example: here are
St. Goar's vacation rentals
There's usually a small cleaning fee that you pay to owners but no agency fee if you go through the owners or through the first website I gave you. Make sure that sheets are included (some owners don't provide them and families bring their own, or they charge a small fee for them.)
German words for rentals: "Ferienwohnung" ("Fewo" for short) or "Ferienhaus".
We've booked rentals directly on numerous occasions with great success. Deposits are not common - I've never paid one. Just book and give your word that you'll show and pay, and that's normally enough.