How would an empiricist develop a moral philosophy without running into the is-ought problem? The Moral Landscape by Sam Harris tries to formulate one but does not effectively resolve the is-ought problem although it does successfully argue against normative moral relativism and express how science and morality can intertwine.
By asking, instead of what is moral, what a person can be persuaded is moral. This is basically an empirical question; we may run experiments where we try to persuade people of different moral positions.
Recognize that it doesn't matter to you what is objectively moral, unless you can also be persuaded that it is moral. If it's "moral" but does not persuade you, then you won't accept it, and if it's not moral but does persuade you, then you will accept it.
You see, what you can be persuaded is moral is more important to you than what is actually moral. (And what I can be persuaded is moral is more important to me than what is actually moral; and what Jane can be persuaded is moral is more important to Jane than what is actually moral.)
This concept is more than simply each person having their own subjective views. A person may hold moral views that they can be argued out of. A person may read a book on Kant and decide the categorical imperative makes sense to them, and thus judge that they were wrong before they had read that book. (Not to single out Kant). What a person can be persuaded of is an aspirational, future concept, not necessarily the same as their views in the present.
Of course, perhaps someone can be persuaded of one moral view, and then persuaded of a different one, and so on. Perhaps they can be persuaded of anything. Given an argument for A, they believe A; given an argument for B, they believe B.
But, as they learn of more and more evidence and argument in favor of different moral positions, perhaps their moral conclusions tend towards some limit. Perhaps, if they consider at the same time an argument for A and an argument for B, they do not accept both A and B; they avoid a contradiction and accept only one, or neither. A reasonable person avoids contradictions. So we might suppose that a reasonable person would eventually converge on some moral position, in the limit as they consider all moral arguments.
We might even have a happy coincidence where any person of sound mind eventually, in the limit as they consider more moral arguments, agrees on some moral proposition. This happy coincidence is a purely empirical situation, but if it holds, there would then be some basis for calling that proposition part of "objective morality."
For my philosophy, Morality is at the central place of any Empiricism.
Human's true understanding is always metaphoric and "correct" within his or her own context at that particular moment when he or she formed such a metaphor, in this deeper philosophical sense, everyone is right and this is exactly the basis of the universal belief: everyone has right to have his or her own belief... But in normal writings, hardly any person will write a sentence so exact and long like a mathematician, that's why I keep saying math is the much-more-clear-while-abstract-dull metaphors for most layman... Numbers like 2 is just a more precise analogy, it's a more abstract-but-more-clear metaphoric analogy of an exact copy of something, but we all agree there're no two exact leaves in this natural world so where did you ever see and experience the concept of "2"? However, "analogy" is also nothing but true "analysis", it works fine in applications and has not be "invalidated" so far as a common metaphoric "description". Literally written as an Arabic symbol, its semantic meaning (more precise common human metaphors) can be limitless, it may be forming 2 apples (object) in your mind, double an existing action (function) in my mind, etc. Our mind just "see" and hopefully "honestly describe" metaphors using other more or less clearer analogies. In this sense, Stoicism which only emphasizes "ethics virtue" is shining here, since the only real job left for us is to "honestly" describe this world full of confused metaphors and ideas, otherwise we're just forever-painfully-confused, indifferently proudly confused, or intentionally deceitful to take advantage of the confused.
Since we cannot have the chance to know the ultimate truth, the only thing left for us human beings are just to "describe" the phenomenal "honestly" to oneself and to others from his or her own experience, otherwise it's like put oil onto fire, make everything much harder to manifest itself and thus be useful. Ethics is nothing but to be honest all the time (faithful to everyone all the time), otherwise no philosophy or true knowledge can be really useful to the dishonest, thus unethical person, because dishonesty magically prevents true understanding of truely honest Saint's words or dogmas, only creates ego, disregard, misunderstanding and distortions... But in reality very few people realized this and act accordingly to this dogma...
You might divide morality and ethics into seperate scientific and engineering domains. Nothing about physics tells you what you ought to do if you would like to build a machine, nor what machines you should build. But if you want to build a machine it informs what you can or cannot do. Nor does any scientific conclusion imply what experiment you ought to perform. You have to have questions and purposes which step behind the next experiment or project. But there are objective confines here.
For an ought or ought-not, we require is/was in conjunction with purpose to explain why you ought or ought not in schematic form. If you include in that "I want to achieve only good purposes," then an empirical sciences of situations morality constrained to Constructor theoretic explanations can inform the behavioural schematic to suggest what you ought or ought-not do. The fact that a shooting was immoral does not imply that shooting will once again be immoral and so you ought not do it. But given knowledge of the conditions and purposes, moral analysis of the history of shootings can suggest your behavioural plan would tend towards good or evil. That analysis might suggest you ought or ought not take the shot.