Actually, I have found that rainy days are the best time to clay your car because you don't have to worry about your car drying out and etching while you are claying.
Don't rely on the rainwater for your lube. After washing & rinsing your car (don't dry yet), clean out your bucket and refill with soap and water and use that as your lube. If you think there may be dirt particles from the rainwater, don't worry because you'll be 're-wiping' each panel with your soap mixture as your lube. When done, pull the car in your garage and dry.
The only downside to this situation is you can't 'sheet' most of the water off during your final rinse. Not really a big deal though...