If the basis of morality is evolutionary and species-specific (for instance, tit for tat behaviour proving reproductively successful for humans; cannibilism proving reproductively successful for arachnids), is it thereby delegitimised? After all, different environmental considerations could have favoured the development of different moral principles.

There's an ambiguity in the words "basis of morality." It might be about the natural history of morality, or it might be about its justification. The problem is that there's no good way to draw conclusions about one from the other. In particular, the history of morality doesn't tell us anything about whether our moral beliefs and practices are legitimate. Even more particularly, the question of how morality figured in reproductive success isn't a question about the correctness of moral conclusions.

Here's a comparison. When we consider a question and try to decide what the best answer is, we rely on a background practice of reasoning. That practice has a natural history. I'd even dare say that reproductive success is part of the story. But whether our reasoning has a natural history and whether a particular way of reasoning is correct are not the same. modus ponens (from "A" and "If A then B," conclude "B") is a correct principle of reasoning whatever the story of how we came to it. On the other hand affirming the consequent (concluding "A" from "B" and "If A then B") is invalid reasoning even if it turns out that often, in typical human circumstances, there's some sort of advantage to reasoning this way. (Reasoning heuristics can be invalid and yet still be useful rules of thumb, though don't bet on this one being a good example.)

I assume the larger point is obvious. We say that stealing is wrong, and there's presumably an origins story about how we came to that principle. But that doesn't give us a reason to doubt that stealing really is wrong.

Not just the same point, but still relevant. There's no such thing as spider morality. Spiders don't subscribe to a code of cannibalism; they just (sometimes) eat their mothers. (BTW: rabbits sometimes eat their young. Happy Easter!) The reason we don't talk about spider mortality is that spiders can't step back and ponder whether it's really okay to eat Momma, but we can. Even if eating your mother might be in your reproductive interest, a little thought should suggest that it's probably still not okay.*

The causal basis of a belief is one thing; the soundness of the belief is another. For some reason, this point often seems less obvious for morality than arithmetic, but holds all the same. Knowing where a belief came from doesn't tell us whether it's true.

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* Not under any circumstances? No. A bit of imagination will let you come up with Bizarro cases where eating dear Momma would be the best thing to do; details left as an exercise for the reader. But this actually reinforces the point. We can reason about what we ought to do. If we couldn't there'd be no such thing as morality. And the conclusions delivered by our reasoning won't always be what you'd expect if you simply looked to evolutionary or social history.