The SEP doesn't say this.

Yes, it does. But it says it in the article Moral Anti-Realism, not the article cited above, Moral Realism. The former article is very interested in objectivity constraints, but expresses a great deal of confusion about how to make sense of them; the latter article mentions them only to toss them out for being too confused. (It would not be too surprising if this has something to do with the latter author being more convinced of the truth of 'realism', hence wanting to make the Realism brand simple, clean, and appealing to a wider audience.)

If your encyclopedia has an 'Apples' article and a 'Non-Apples' article, and the two articles completely disagree about what it means to be an 'Apple', then you have your first clue that the word 'Apple' should always come pre-tabooed.

(ETA: More generally, be aware that 'the SEP says X' is less reliable than 'SEP article Y says X', because articles may disagree with each other. SEP is an anthology of introductory essays. We wouldn't normally say 'Very Short Introductions says X', even if we trust the VSI brand quite a bit.)

What the SEP actually says is, "Moral realists are those who think that, in these respects, things should be taken at face value—moral claims do purport to report facts and are true if they get the facts right," and that's it.

Almost. Moral realists (even on the more inclusive definitions) also demand that at least one moral claim of this sort be true. (This is asserted in the sentence right after your quotation terminates.) That's why error theory is not a form of moral realism; realism is a (perhaps improper) subset of success theory.