I don't think there are any anonymous data sets of this nature. There are, however, a number of published feature analyses of LMS software, including many in peer reviewed journals. Try searching Google Scholar with keywords like "Moodle feature analysis" or "lms feature analysis" -- you'll find several institutions who have published the process they used to select an LMS.
You may be able to construct a data set by reviewing these publications. I think you will find that the criteria used to select an LMS vary widely between different types of institutions, and some institutions seem to assemble a huge list of all the features mentioned in any LMS marketing literature and then look for the product with the most features (which isn't a very good way to choose a system). Without knowing more about your research question, I can't advise further.
Good luck with your research!