The scientific enterprise works because two claims are true:
(1) Our external-facing senses match reality more enough of the time.
(2) Enough people can intersubjectively agree upon their models of reality.
These may be necessary but not sufficient, but I don't think that is a problem. Note that people unable to or unwililng to do science do not disprove the claim that there exists an objective reality which can be increasingly well-known.
I claim that there exists an analogous situation vis-à-vis morality:
(1′) Our internal-facing senses match morality more enough of the time.
(2′) Enough people can intersubjectively agree upon their models of morality.
Eric Schwitzgebel's 2008 The Unreliability of Naive Introspection is an important work, as it supports the claim that just as our external-facing senses need a lot of 'tuning' before they can be called 'reliable', our internal-facing senses (i.e. 'introspection') also need a lot of 'tuning' before they can be called 'reliable'. Science has demonstrated that people can sufficiently circumvent their own cognitive biases, especially when they work together. But does there exist an 'internal order' to the human psyche, which is comparable to the observed order of external reality? I claim that there is law-like behavior of humans even when the label 'irrational' is used; else the book Predictably Irrational would not be possible. My answer to 'What arguments have there been on the relative merits of praxeology vs rationality?' may be of interest, here.
An argument against (2′) is that there is exists a terrible amount of disagreement over how people ought to behave. I propose that this is because much 'research'/discussion of morality has been verificationist, where we ought to at least have included a much larger component of falsification. Compare the two claims:
(A) X is a good way to treat people.
(B) I feel hurt by the application of X.
Do not be misled by the word 'feel'; the introspective words are inherently more subjective-sounding than the words we use to talk about the perceptions our external senses deliver. I claim that too much application of (A)-type thinking fails to construct hypotheses which are allowed to be falsified. If the analogy,
external-facing senses : reality :: internal-facing senses : morality
holds, then we should expect objective morality to be researchable, just like objective morality. The term 'research' connotes "a hypothesis might be wrong". Claims which might be wrong must have established tests for knowing if and when they are wrong. Failure to properly utilize (B) would hinder research; it would be a form of confirmation bias.
I define 'objective morality' as the demarcation line between:
(I) Behavior which brings about pleasure/happiness/etc.
(II) Behavior which brings about suffering/sadness/etc.
It is an open debate as to whether there is a way of believing/behaving which would allow every person to experience (I) more and more, and (II) less and less. But even this does not falsify the existence of an objective morality, because there exist people who cannot do science, and that does not falsify the existence of an objective reality. See the accepted answer to 'Does the existence of psychopaths pose a problem for moral realists who argue we have a moral intuition?'