The answer to that is kind of tricky. I haven't found any literature directly discussing the relation between non-profit structures and market share, but here are a few ideas.
Usually for-profit companies try to maximize (surprise!) their profit-function in the long-run, even though they might focus on other stuff like market share or prestige in the short- or even medium-term. That is because shareholders in for-profit companies (for the sake of argument, let's define 'shareholders' more broadly than in the conventional sense) try to maximize their own consumption as well, meaning they will try to raise they're budgets to elevate their possible consumption, subsequently increasing their utility.
Non-profits on the other hand focus on some other thing to maximize. For the most part, their goals are idealistically driven, for example minimizing global CO2 output, or maximizing survivability rates of marine life.
It wouldn't make sense for a nutrition company to focus on maximizing market sharewhich would lead to a noticably increased presence on the shelves, without returning to a profit-maximization routine in the long-run.
Keep in mind that in economics, monetary value is (beside other functions like driving down transaction costs) just a way to approximate a certain good's real value. Cost-minimzation, which is typically done within a profit-maximization routine, could be seen as a constant re-evaluation of the value of goods used for producing something else.