Is there a branch of sociology that deals with nutrition and active lifestyles?
There is a branch of sociology, which would be the sociology of health. Research into 'active' lifestyles and nutrition would fall under sociology of health.

Unlike public health which will take a much more straight forward approach in addressing issues based on biological/essentialist research, sociology of health will look into the social constructs of health, how that's implemented in society and the issues surrounding it (i.e. fat discrimination, issues of class/race/gender in the 'health' rhetoric' and so on.

Many universities with sociology programs might offer an undergraduate course in the sociology of health. I took one and loved it. It's not my main focus in sociology but I do find the social construct of health and medicine highly fascinating, especially in the way in seems to be used to justify our discriminations.