In addition to cardsorting.net (formerly known as MOIST), there's also Qualtrics (a platform that many academics use) and OptimalSort (formerly Websort.net, just purchased by PWC). Here's an honest breakdown of the relevant features:
Let's take an example. Assume that you're sorting potato chips, and that the next is Lay's BBQ kettle cooked chips.
You could see it as belonging to a pile of BBQ chips (with Ruffles BBQ), to a pile of Lay's chips (only Lay's) or to a "healthy" pile. Makes sense?
In a single-card sort, there's only one card per item. The participant must choose which it fits best, and put it into that pile. If you sort the chips in your BBQ pile, does it mean you do not see Lay's Kettle Cooked as healthy? Or it's somehow different, in your mind, from all other Lay's chips? Of course not. Yet, in a single-card sort that's what your data suggests.
Are USB memory sticks only in one category of BestBuy's website? Are bananas only located in one section of your favorite grocery store? Most of my research in psychometrics and marketing has been about producing statistical models that can handle multiple sorts data. There are also white papers (see resources tab) that illustrate the simplistic (and often wrong) inferences when single-card sort is analyzed.
First, your can cite my article multiple cards sorting task that appeared in Psychometrika (link here), the world's leading publication on the development of quantitative models of psychological phenomena. Second, PhD thesis (link here) would be a good one too. That should be a good start.
Reason 1: Card sorting (aka the sorting task) has a long and rich history, particularly among psychologists. These psychologists gave participants one card per object sorted. The requirement to have only one card per item was partly due to logistics of handling actual paper cards. It was also partly due to the fact that in most psychological experiments there was only one way to "correctly" sort the items. One card per item made sense then. But not now. Why limited yourself to one card per item?
Reason 2: There was no web interface that allowed multiple cards per item. Now there is!
In a closed-sort, the piles (categories) are provided by the researcher. Participants then assign items to the piles where they best fit. It is considered to be more confirmatory than exploratory.
In an open-sort, the piles (categories) are notprovided by the researcher. Participants simultaneously determine 1) the number of piles that they want, and 2) the assignment of the items to their piles. It is considered to be more exploratory than confirmatory.
It's free. Why would you?