Rain barrels are great. At my old house I had about 2,000 gallons of capacity thanks to some rather large IBC totes in system. That being said, I see two issues with your potential setup. First, raising up the rain barrels adds roughly half a PSI of pressure per foot raised. That means unless you are going to be using a pump you'll be using 1-3 PSI water. That's probably not enough to run the current system. The second issue is that the water you collect isn't going to be nearly as clean as the water supplied by the city. Your plants won't mind the pollen grains and other stuff in the water, but it can be an issue when trying to push water through a small opening like an irrigation emitter.