It occurred to me that I could try steadying the camera on something, so I leaned my elbows against the car and tried again. Still not a great shot, but there was some improvement.
Finally, I rested the camera on the roof of my car and hit the shutter button. It took a couple tries to actually get the moon in the shot, but eventually I did it. I still think a tripod is in order, but I was encouraged enough to stay outside practicing until my fingers got too numb to handle the camera. Here was my last attempt. I love the way the branches of this locust tree look when they're bare.