DIstilled water is not healthy to drink. It depletes the body of a lot of things including calcium so should not be used in a mister, which is the water your chameleon drinks. Do a google search on why you shouldn't drink distilled water.
I use RO water. At the moment I'm in the middle of researching a RO system so still haven't one in my house. I buy my RO water from a filling station outside the grocery store for $1.50 for five gallons. I have a large collection, going through at most 15 gallons a day, usually less.
I would suggest you simply buy RO water in three or five gallon jugs and have them refilled as I do at a coin-operated water filling station. Don't put all the water into the reservoir bucket since you want a fairly quick turnover of the water and not allow it to stand in an unsealed container open to dust and contaminants. When I find my nozzles are not working as they should, I run a few gallons of distilled water through the system and that has cleaned up anything that has clogged the screen.
If it is safe for you to drink the water, it is safe for your chameleon. I find it a little odd that people are more concerned about the safety of tested municipal water for their chameleon than they are for themselves and their children, Flint Michigan aside. While Flint showed some serious problems with the testing and the timely response of those in charge of protecting public safety, in general, municipal water is safe to drink. Honestly, I wouldn't worry about my pet chameleons long-term health even if I had been giving them water from the epicenter of the Flint crisis. Most chameleons have very short life spans even with the best of care, although I might be a little concerned about my breeding stock, though.