Normative Ethics is often described as prescriptive. Do the questions posed by these ethical theories apply in moral anti-realist frameworks?
Yes, in general non-realist frameworks for understanding ethics include efforts to make sense of ethics. At minimum, they might try to account for what’s going on when we use moral language. Often they will try to account for why human beings generally care so much about rightness and wrongness and other moral statuses.
So, for example, in Ayer’s radical, early 20th century emotivism, the symbol “is right” is taken to be “cognitively meaningless” because it cannot be shown to be true or false by any possible observation. Comments about what’s right become merely expressions of feeling. But that doesn’t make them meaningless. Even if moral claims merely express our feelings, we can still meaningfully try to persuade others to share our feelings, feel more strongly about some things than others, and—to your question—ask others whether they feel the same way and why. All of that can be done despite that there aren’t discoverable, scientific facts about rightness.
More current non-realist metaethical frameworks like Blackburn’s quasi-realism and Gibbard’s expressivism similarly account for what’s going on when we debate moral questions, though their accounts are quite a bit more sophisticated than Ayer’s.
In general, though, we can substantially disagree, and even offer reasons, about something without requiring that it in the end be a matter of fact or independent existence. We can debate who’s a nerd, for instance without committing to nerdiness being a scientifically-real human-independent phenomenon in the universe nor even that it is anything like precisely defined. Same for ethics.
Moral objectivity, hence normative ethics (an ethics of moral prescription), does not depend on realism. Moral realism assumes, on the most common interpretation, (1) that there is a way the world is morally, specifically that it includes moral properties, that does not depend on what we say, think or believe about it; and (2) that we can and to some extent do know how the world is as regards these properties.
If anyone presents an objectivist ethics, it is Kant : but Kantian moral objectivity is unconnected with moral realism. For Kant, morality is a requirement of our rational nature and our capacity to act on consistent rules binding on all human agents qua rational. And Kantian ethics is prescriptive ('Always treat humanity, whether in your own person or in that of another, as an end and never merely as a means') in what in terms of your question is an anti-realist conceptual framework. Anti-realist because knowledge of moral properties is alien to this account. So yes : normative ethics is perfectly possible within a moral anti-realist framework.
For the record I understand metaethics and normative ethics to be in quite different lines of business. Normative ethics is concerned with e.g. the permissibility of euthanasia, whether there can be a Just War and if so under what conditions, whether we have moral obligations to future generations and so on. Metaethics is occupied with the analysis of the concepts and logic involved in such issues and in the moral judgements made about them. How do we unpack - enumerate and specify the components of - the concept of moral obligation, for instance ? And is it logically possible to have a moral obligation to the non-existent, as future generations are ? Metaethics and normative ethics are different inquiries even though normative ethics supplies most if not all the data on which metaethics works.
Your own question is metaethical because it asks a logical question about normative ethics, namely whether its prescriptions apply - are logically capable of applying - in moral anti-realist frameworks.