No, they will drink tap water. I would think it would have to be quite awful tap water for them not to.
If it is that bad, you definitely need to figure out another water source.
Heres the thing.
Yes, the chameleon is supposed to get wet during misting sessions. What you want to do, is simulate rain. It may take 10 min for the animal to start drinking, but it will drink if it is continuously raining in its environment.
You want sessions at least 5min long, 3-4 times a day. Allow enough time in between for the cage to dry out though. This will greatly reduce any chances of mold or mildew developing in the enclosure. You especially want enough time for the enclosure to dry out before lights off.
You want at least one 10 min session a day as well.
Its a good idea to give them a 30min+ session once every week or two as well. Taking them outside, or placing them in the shower if necessary.
What are you using to "mist"?
The term misting can be misleading. Mist is great, it will cover more surface area with water, and help raise humidity. It will cool the enclosure down. What it wont do, or iss not likely to do, is provoke a chameleon to drink.
Think rain.
Placing a lot of foliage in the cage(you should really use live plants, especially with an omnivore) will collect the "mist" and it will "rain" down from the foliage. So having some high up in the enclosure, with an open space below that has branches/vine running through it is a good idea.
The chameleon will have a semiprotected area to relax, and drink its fill during these sessions.
An automated misting system is great, its a life saver. However, I firmly believe all keepers should have one of these bad boys:
http://www.lowes.com/pd_112857-1306-190377_0__?productId=4193785&Ntt=pesticide+pump+sprayer&pl=1¤tURL=%3FNtt%3Dpesticide%2Bpump%2Bsprayer&facetInfo=
As nothing is as good as a well executed hand misting session now and again.
You indeed also want a dripper going, anytime lights are on.