Were any crusades in particular more receptive to woman’s involvement? What are the best primary sources to learn more about this from the time?
This is a relatively new and under-studied topic in the history of the crusades, so there is always new work being done and much more can still be said. But I have some previous answers that may be helpful:
What were women's roles during the Crusades?
How was the life for a crusader's wife in and outside the Holyland?