I have been watering my plants with tap water ever since I've been growing peppers (and sunflowers) which, to be honest, has really not been that long so far (just a couple years). I have yet to see a water-related problem that wasn't under- or over-watering. I mean, these are peppers we're talking about; if it were some sort of carnivorous plant or something which has much stricter mineral and water additive requirements, then yeah--distilled water or water that has went through reverse osmosis would be the way to go, if no natural rainwater is available. And yes, my water contains chlorine. In fact, with a quick Google search recently I found this interesting info just recently:
http://www.google.com/url?sa=t&source=web&cd=2&ved=0CBcQFjAB&url=http%3A%2F%2Fwww.cityofalliance.com%2FWaterQualityReport.pdf&ei=yDxFTIqrCsK88gbO8PQN&usg=AFQjCNFdLUwdsZfO-zA117Q4LYpCWSIm3Q&sig2=ZhVTlQ30r_F6g0inOQqm0Q
Yep, a PDF describing my city's water for the last few decades, what it's like now, and where the city is seeking to improve in the future. I simply searched for "Alliance, Ohio water quality". Maybe you can find out what your local water contains with a similar search? But really, I don't think pepper plants are that picky if their water contains low amounts of mineral salts and other additives such as chlorine. Whether in the garden or in pots, my plants have definitely had their share of tap water with no directly-resulting ill effects. Hell, apparently my water even contains fluoride to promote human dental health, and the peppers don't seem to care; they're growing great.
On the other hand, I refuse to attempt to water my Venus Flytrap with city water; their soil must be acidic and completely devoid of any nutrients or mineral salts whatsoever in order to avoid damaging (and eventually killing) the plant. But the flytrap is a special case; it really is picky about its soil. Too many nutrients from the soil is deadly... they were evolved to get nutrients from catching prey. In my experience and from what I've read, peppers are nowhere near as picky.