Many readers may expect that an "-ism" refers to a belief, as in "fundamentalism", "theism", or "Darwinism". However, "-ism" can also refer to a social institution or practice, as in "capitalism" or "communism". A capitalist economy isn't just one whose participants have capitalist beliefs — it's an economy that is structured in a particular way, with people actually playing the economic roles of investor, entrepreneur, employee, etc.

Similarly, terms such as "sexism" or "racism" can refer not only to biased beliefs about sex or race, but to sorts of social institutions in which some people exercise political power over others on the basis of their sex or race. Since there is a clear answer to the question, "Historically, which sex has exercised political power over the other in human culture?" the question "Who can be sexist?" seems to be dissolved and there is no need to argue definitions.

(Yes, some folks do use "sexist" as a near-synonym for "evil", just as some libertarians use "socialist" as a near-synonym for "evil" — I've heard it asserted that monarchy is "socialist" because it doesn't respect individual liberty, for instance. But we don't have to take that kind of silliness seriously.)