Do confidence intervals apply to quota sampling? French polling institutes are currently facing a major crisis after they recently published what can only be called the most ridiculous poll so far on the 2012 presidential election horse race. The French Senate is now considering to legislate on the issue by forcing polling institutes to publish, among other things, the confidence intervals for their results.
However, some pollsters are opposing the measure, claiming that confidence intervals do not apply to quota sampling, which is the method used by polling institutes in France. Since quota sampling is formally non-probabilistic sampling, there is some truth to the claim. But since quota sampling is fundamentally stratified sampling, confidence intervals should apply, right?
May I ask for experiences about this issue outside of France, in countries where pollsters also use quota sampling?
 A: As whuber says, the short answer is that quota samples are the "poster child for outmoded, known-bad sampling methods" and "have long been discredited."  The longer answer is that there may be conditions under which "quota-like" samples can work reasonably well.
Exhibit A here is recent work on reconstructing representative results from opt-in Internet panels. This paper gives the statistical grounding for this approach.  To make a long story short, typical sampling schemes 1) draw a random sample, 2) attempt to recruit subjects, and then 3) add post-stratification weights to compensate for differences in who responds.  In the opt-in approach, you 1) recruit subjects non-randomly, 2) compare responses to a representative baseline, and 3) add weights to compensate for the differences.
In terms of practice, opt-in sampling is similar to quota sampling, but the statistical foundation is more developed. The upside is that you can make claims about representative sampling, confidence intervals, etc. The downside is that your claims are based on difficult-to-verify assumptions about how people self-select into your sample.
A lot of people are skeptical about these methods -- they sound too much like quota sampling. But some evidence suggests that opt-in sampling can work well at least some of the time.  So despite the controversy, Polimetrix/YouGov (an early adopter of the opt-in sampling model) seems to be doing reasonably well.  Among other things, they've done all the data collection for the Cooperative Congressional Election Study, a series of recent academic U.S. national election studies.
(I'm pretty sure ICPSR carries this data.  If not, Harvard's social science dataverse certainly does. Lots of academics are using data from these samples.)
Anyway, you asked about quota sampling.  As you can see already in the comment thread here, any well-trained pollster will tell you that quota sampling is bunk. The jury is still out on opt-in sampling. For the time being, if you want to draw confidence intervals around quota samples, I'd say these methods are your best bet.
A: *

*In most non-compulsory survey contexts, there is a substantial problem with nonresponse. This from 2002: "the recently reported estimate of survey cooperation rates from CMOR, the Council for Market and Opinion Research [USA], averaged only 14.7 percent." and from Paul Gerhold, "I believe that it is still possible to draw random samples. I just don't believe that it is possible to execute them." In this context, the fact that the SAMPLE is random isn't very relevant, because the resulting data isn't.  

*This makes bias adjustment the major issue in valid estimation, and field method design is an important component. The ways in which one might want to do this, and the resulting confidence estimates, are well beyond what can be discussed here.
