The closest kayaking to Flagstaff is Oak Creek. A couple Ark Valley crews went down last March for a record runoff...
The upper reaches contain some V+ creeking that only runs after heavy rain...then comes the Slide Rock section which contains predominately III+ to IV+ whitewater with 3 solid V's:
Below this is the Indian Gardens section, which is rated as solid IV. This stretch ends at Sedona, and marks the beginning of more mellow water.
The Sedona to Red Rock stretch contains predominately class II whitewater with a half dozen III+ to IV- moves depending on wood and flow:
The Red Rock to Paige Springs run looked to be clogged with wood, so we didn't run it. However, you can put in at Paige Springs and float down to Cornville. This stretch contains more class II water with a handfull of easy III's.
When we did it Oak Creek was running 650 at Sedona for the upper stretches and 850 at Sedona for the lower.
Here is the gauge: USGS Real-Time Water Data for USGS 09504420 OAK CREEK NEAR SEDONA, AZ
It is a really fun creek, but doesn't run every year. Hope that helps!