We plan to distribute the survey via platforms that users are already using to learn about themselves: particularly, via Buzzfeed quizzes. If the brand has a mailing list we will distribute there as well. This might seem like a biased sample but it actually serves us well — the opinion of someone with no interest in fashion or no chance of ever buying from the brand are of little value.
As a reward for the time spent taking the survey, survey-takers will receive a free phone or desktop background download based on how they answered the questions in the survey — an incredibly inexpensive but desirable incentive.
Here is an example of a phone background for a survey distributed by Kate Spade:

And another for ban.do shoppers:

Notice that each background contains a small link in the corner. Ideally, the survey will be a self-catalyzing process: we believe that other fashion-minded friends may ask their friends who have already taken the survey where they got their phone and desktop backgrounds, causing friends to organically share the survey. We probably need to experiment a bit more to figure out how to make the survey link unobtrusive but still usable.
As added incentive, 1 in 10 people will get to meet the CEO of the company at a pop-up store. This will allow for both a breadth of information collected via the survey, and a depth of information collected via 1-on-1 conversations with shoppers at the pop-up stores. Seeing items in person will also presumably allow for deeper information-gathering because shoppers will be able to see and even touch clothing items that they had previously only seen in small thumbnails.