Many people turn to science to answer the question, however "science" cannot. By the term "science" is typically meant what can be quantified in laboratories or by laboratory equipment in the field. The soul has not been detected by those means. It is not for a lack of trying though. Many attempts were made to weigh the body near the point of death to detect a decrease in weight that might be associated with the departure of the soul. There are countless other attempts, but usually with the intended purpose of proving there is no soul. At the end of the day (I have been wanting to use that expression since it became widely used.), science has no data on which to speak to the existence, beginning, departure, end, or any other properties a soul might have.
This was actually noticed quite long before the world developed much "science." Ancient religious and philosophical texts describe how there is no circuitous route to detect the soul (or various similar phenomena) and that it is only perceived directly. That is, the soul can be self aware. The "I am," the "Atman," the "self," and other similar phenomena can remain beyond the reach of material instrumentation.
It should be noted here that it can be possible using scientific methods to establish certain instances of "spiritual" phenomena (sometimes called "proofs of the spirit"). The results however are typically not repeatable and therefore not accepted by people without first hand experience. "You had to be there," can be a big problem.
Although "scientists" typically avoid discussing souls, at least in their role as scientists (because of the laboratory requirement), most religions and rather well respected philosophical systems do examine at great length the likely properties of souls. Much of that is because science is so limited in ordering society. When everyone agrees what the problem is, science can often find answers. When people do not agree what the problem is, which is typical in politics, science might be useless to solve anything.
Judaism and Christianity recognize a phenomenon that is "not of this world" and that the true self is merely "hosting" the temporal body. Religion has more and better tools for the ordering of society than science has. While in the United States religion may not dictate state policy, it may advise it.
Returning to the question likely important in discussions of when "life" begins in the womb, when might the "soul" enter the developing body? Notice that "science" has nothing to say. What then has religion to say? Nowhere in the Bible is it specifically expressed when the "soul" enters the developing body. A scripture that is often cited in these discussions is Jeremiah c1 v5, "Before I formed thee [Jeremiah] in the belly I knew thee; and before thou camest forth out of the womb I sanctified thee, and I ordained thee a prophet unto the nations." While that verse does describe some sort of early beginnings it is not clear what existed "before" entering the "belly" or when it entered. It might even be different for different people and their circumstances.
The notion that "life begins at conception" can make more sense if the existence of the soul is ignored. At conception the DNA "program" for that individual human is assembled. This is usually accepted as science. It is very much like a "computer program" and runs things thereafter — for the body at least. Computers don't have souls, although "Short Circuit" is still a good movie. There is quite much in literature and entertainment on "the man in the machine."
When and how the soul and body interact is a matter of speculation. As generally understood, Darwinism does not accept the Lamarckian notion of gene (or program) modification by the life experiences of the organism. However, the exact nature of "instinctual" behavior or "genetic memory" can introduce questions that might require answers beyond Darwinism.
Abortion laws must balance the rights of the mother with whatever other rights they decide to introduce. This is especially true if the rights they intend to introduce have no sound basis. A tragic mistake of the Supreme Court justices in the Dobbs decision was to summarily disregard the rights likely intended for women too in numerous parts of the Constitution, especially the 14th amendment. They have gotten away with it so far because the people who wrote those parts of the Constitution are no longer around to tell them how wrong they are. An obvious solution is to amend the federal Constitution, or perhaps several state constitutions first, to establish certain rights for women. Remember, the Supreme Court does not, may not, "write" any parts of the Constitution, it may only "interpret" them. If amendments to state constitutions "written" by the people are ratified that make it clear that women have rights to "life, liberty and the pursuit of happiness" for example, the courts may not "interpret" those rights away. The people who wrote those rights can stand directly in front of them and tell them exactly what they mean. Because of the obvious blunder of the Supreme Court that will likely best be done at the state level first.
When the rights of women are clarified and guaranteed, certain types of abortions might still be banned anyway. The more ridiculous bans might not stand though. The important thing is that the rights of women are included in all the calculations.
A hazard in setting some arbitrary definition of the beginning of "life" at very early stages is that it can lead the mother to a "rush to judgment." That is, under the pressure to decide early what to do she might more likely choose to abort than she would have given more time.
In their desperate attempts to seem competent the Trump base looks to "overturning Roe v. Wade" as some grand accomplishment. It does sound impressive. However, long experience in trying to restrain abortion by means of government has proved to be less than successful even before Roe v. Wade, especially before Roe v. Wade. Some things are simply beyond the purview of government, and will likely remain that way.