I would highly recommend Canyons Incorporated out of McCall Idaho. They do 6 day Main Salmon trips. I have done many trips with them and know most of their guides very well. All of their guides are long time veteran river runners, not just 18-20 year old kids looking for a summer job. These guys are pros, and they have about an equal number female guides as male. If you want a top notch river experience, I'd contact them. As Dave Frank posted.... the Main Salmon is a perfect family trip river. Canyons River Company | Salmon River Rafting | Middle Fork river Rafting
As others have said......licensed guides don't usually have their own commercial permits to do trips separate from the company they work for. Buying out a commercial trip for just your family would be expensive. In my opinion, meeting new folks on a river trip is part of the fun, so just go with the regular group. Your kids will probably like that more.
Your odds of hooking up with a private group during the dates you want would be next to impossible. Plus you have no control over who these folks are, what their river style is (family vs party folks), what their experience & safety level is. Private trips expect all trip members to pitch in and do equal amounts of work (loading & unloading boats & gear, cooking, clean-up, groover duty, etc.). If you pay a private boater money to do these things for you, you've made the private trip illegal, and put the trip leader in jeopardy of big trouble.