The USGS fact sheet listing volcanoes to climb and picnic on, lists Arizona's Sunset Crater at the top of its list, but then notes it's a national monument and you can't climb on it. [maybe that's why the link is broken?] But, they point out, there are lots of nearby cinder cones in the San Francisco volcanic field that you can climb. They list 20 volcanoes to climb in the western U.S.
[right, SP Crater, one of the other cinder cones in the San Francisco volcanic field. Credit, USGS]