Reddit reviews Body & Soul: Notebooks of an Apprentice Boxer

Yes you can make a sociological project out of martial arts. Loïc Wacquant's ethnography of boxing in Chicago's South Side comes to mind as an example of such a project.

Here is a literature review on the sociology of combat sports that may be a good starting point for your project. The authors of this review have a whole section on "body pedagogies" which seems very relevant to your interest in the intersection between education and martial arts.

You may also be interested in this collection of ethnographic research on martial arts.

Loic Wacquant, Body & Soul: Notebooks of an Apprentice Boxer. Such a great example of deploying Pierre Bourdieu in ethnographic fieldwork book!