According to my professor, Sharon Street is both a cognitivist and a moral anti-realist. Are the two compatible? How can a statement be truth-apt if there is nothing in the fabric of the world that fixes its truth value?
I think you're winding up feeling this conundrum because you may be confusing moral anti-realism with a denial of the existence of moral claims in our discourse. I can't remember the last time I read an article by Sharon Street, so I cannot comment on the truth of the claim in question.
Cognitivism, in this context, is the view that there is a truth value associated with moral claims (http://plato.stanford.edu/entries/moral-cognitivism/#Cog).
Moral anti-realism is not as easy to define as it sounds (http://plato.stanford.edu/entries/moral-anti-realism/ N.b., I would also recommend the article itself for help in clearing up this issue). At least part of it seems to be some form of negation of moral realism. One species is to deny that the things we say about morality are about some knowable thing called morality (this would lead to moral non-cognitivism). Another species, however, is to deny that that the things we say in morality follow from things themselves (this is the species called there non-objectivism).
According to the article, Street is a type of constructivist. This is to say that on her view, there are moral frameworks and they do something, but they are not inherently reflective of anything like a natural law or moral order that is pre-built into nature. Thus, the claims of these moral systems do have truth values, but they have them in light of the constructed "moral game".
Categorizing constructivists is messy business, because there are some constructivists who see the shape and form of the moral system we construct as a necessary consequence of something in our constitution, whether this is a basic evolutionary pathway or, as Christine Korsgaard believes, a function of rational agency. (She thinks that being a rational agent generates the system of morality and obligates all such agents -- she also thinks this is what Kant thinks).
To summarize, cognitivism is compatible with some definitions of moral anti-realism. And this is for the species where the claim relates to the metaphysics of moral claims and their grounding out there "in the world."