I second that the Allan book is the best guide to camping trips in Oregon. However it is far from extensive. The web and some BLM guides give you some real great info. I'm not familiar with Eastern WA, but as far as Oregon is concerned, East of the Cascades you have some great options for multidays with kids.
The easiest would be the John Day River. There is nice 69 mi float from Clarno to Cottonwood Bridge. The JDR is undamed but should still be flowing in early summer. This stretch is almost entirely class II except one easy class III a few miles below the put in. This stretch is the most scenic and remote part of the river. And the access roads are all paved. A shorter and easier stretch exists upstream. A nice scenic stretch runs from Burnt Ranch to Clarno. But if the kids are looking forward to whitewater this isn't the place. I have no experience with the NF John Day so I'll leave that to others.
For the most action I would check out the Deschutes. It is very forgiving class III and mostly safe pool-drop rapids. There are 2 good camping options. First is putting in at Trout Creek. The only rapid of significance is Whitehorse. Most people run it right. You can run it either way but personally I think running left is easier and much safer if you're wanting to make sure everyone stays in the boat. If I was running this stretch with younger kids I would take out at Harpham Flats. If you want more action you could take out in Maupin or Sandy Beach in order to hit the best rapids on the Deschutes. Deschutes option #2 is from Mack's Canyon to Heritage Landing. This section has a few bigger volume class III's but they are pretty much read and run, hit the wave/hole straight on type of rapids. The Deschutes is a mixed bag for camping trip. It has fun but safe rapids and pit toilets at most campsites. But wilderness camping it is not. It is popular with fishermen and floaters due to its proximity to Portland. And a train track runs the length of both of these sections that will probably wake you up at least once a night.
The Grand Rhonde is a beautiful stretch of river with mild to moderate whitewater that should be flowing most of the summer. The season lasts longer then the John Day. Not as fun of rapids as the Deschutes but much more remote and scenic, and nice camp spots.
Another option is the Snake River through Hells Canyon. It makes for a long shuttle but if you put in at Pittsburg Landing and go to Heller bar you will avoid all the knarly class IV stuff above, but still float through a very cool canyon. This would be big water class II, a little III ish with some fun wave trains. There are some nice campsites. They get nice and sandy below the Main Salmon confluence. Very family friendly stretch of river.
If I were looking to take my family on an introductory multi day trip in the NW but wanted to still keep it safe for the 7yo I would probably either pick the Grand Rhonde or Hells starting at Pittsburg.
A great site for info on many of these river trips is
Whitewater Rafting Campsites | Resources for Planning Rafting Trips