Morality is subjective, so veganism can't be objectively right
If that holds, it holds for every moral claim including the ones the speaker relies on. And most people already accept that causing avoidable suffering for pleasure is wrong.
There is no moral fact built into the universe; moral views are expressions of human feeling, culture and upbringing, and people of sound mind disagree deeply about where animals stand. Since no one can show that their values are true rather than merely held, vegans have no standing to call eating animals wrong in a way that binds those who do not share their values, and treating their conviction as an obligation on everyone else claims an authority that no moral view possesses.
Meta-ethical anti-realism is a respectable position with serious defenders. The difficulty is that it is almost never applied consistently when raised here.
If no moral claim is objectively true, then neither is the claim that it is permissible to eat animals, nor the condemnation of anything the speaker does object to. Scepticism applied to everything decides nothing between options; applied to one conclusion only, it is not scepticism but a preference. In practice the argument is rarely a considered anti-realism and more often a way of declining the question. Moreover, veganism does not need moral realism: it needs only the near-universally held intuition that causing avoidable suffering for minor benefit is wrong, which the speaker almost certainly holds already.
Applying a principle generally while claiming an exemption for one's own case, without giving a relevant reason for the exception.
How it applies here: Moral scepticism is deployed against this conclusion while the speaker's other moral commitments are left standing. A general scepticism would dissolve those too.
Introducing a point that is not relevant to the question under discussion, drawing attention away from it rather than answering it.
How it applies here: Meta-ethical scepticism, if accepted, undercuts all moral claims equally and so provides no reason to favour one course of action over another.