This is my blanket disclaimer that "Native Americans" encompass diverse cultures across two continents and thousands of years, so there is not and never will be a singular answer to a question like this.
That said, you may be interested in my previous comment briefly touching on Aztec midwifery, reproduced here for convenience.
The problem here is that while we have definitive evidence that abortion was known and practiced among the Aztecs, the details are a bit hazy. Women's issue, in general, tend to get short shrift in historical works. Something like abortion, which was a crime among both the Aztecs and the Spanish, doubly so. That most of the early Spanish chroniclers were friars does not help either. So I could rattle off a dozens of different herbs that Sahagun's General History of the Things of New Spain says can help with everything from gout to fever to diarrhea to UTIs, but the primary sources are mum on the specifics of abortifacients.
What we do have though, is some vague mentions that midwives would use "herbs" to induce abortions. There are a several plants with abortifacient properties in the Aztec pharmacopoeia, though recorded for other properties. Some of these are still used in Mexican folk medicine to terminate pregnancies, such as avocado leaf or aztec sweet herb. What has been most focused on though, is cihuapatli, which literally translates as "women's medicine."
The focus on cihuapatli is because it is explicitly named as an obstetric medicine. Its primary indication was to induce/hasten labor in full term women. As the Badianus Manuscript -- a collection of herbal remedies published in 1552 -- puts it:
If the woman suffers difficulty in the bringing forth, then that she may give forth the feotus with little effort, let her drink medicines from the bark of the tree quauhalahuac and the plant cihuapatli, the small stone eztetl, and the tail of the small animal called tlaquatzin [opossum].
Quauhalahuac is not identified and eztetl stones were put in just about every concoction. One candidate for the cihuapatli, however, was tested in modern times an found to have oxytocic effects. If you're not up on your obstetrics, oxytocin is the key hormone for regulating labor, as it leads to cervical ripening and uterine contractions which together expel the fetus. Though ostensibly used only for labor difficulties, its mechanism of action could also work to terminate a pregnancy. Medical abortions today use misoprostol for the same oxytocic effect. As for the possum tails, Ortiz de Montellano (1990) posits that they may have an unusually high concentration of prostaglandin, which similarly has oxytocic effects. Nature is weird.
As for how to procure such a thing, we return to the aforementioned midwives. They were a recognized profession and a type of ticitl (physician). More than just attending the birth, they were involved from the very start of the pregnancy doing everything from warding against bad omen and behaviors (like chewing chicle) to providing massages to administering medicines for anything that might ail the pregnant woman. If anyone would have access to abortifacients, it would be the midwives.
Again, though, abortion was highly illegal among the Aztecs. Somewhat ironically, having an abortion or helping someone have an abortion could be a capital offense. A woman's role as a mother was a central to Aztec life as a man's role as a soldier. It is not a coincidence that men who died in battle or were sacrificed, and women who died in labor shared the same afterlife. Because of this prohibition against abortion, and historical bias, our scenario for how specifically an abortion would be carried out would have to be hypothetical. We could speculate that a rich woman living in Tenochtitlan might have more opportunity to employ midwives and greater access to specific herbs due to the central role of Tenochtitlan as a trade hub, but this would just be speculation. We do know that specific plants were normally used in a way that could produce abortions, and that there were other plants in use that we know have abortifacient effects, but the specifics are harder to come by.
I'll be relying heavily on Barbara Alice Mann's Iroquoian Women for this. Chapter 5 of that book contains a lengthy discussion of abortion among pre-Revolution and post-Revolution Iroquois women.
Prior to the American Revolution, abortion would have been a common and accepted aspect of Iroquioan society. Abortions performed early in gestation relied on decoctions made from the bark of American beech or the toothache tree; those performed later used redroot instead.
As early as 1703, we have written records (via Lahontan) of Iroquoian women using herbal abortifacients and contraceptives. In 1782, Thomas Jefferson wrote on the topic as well - though about Native women more generally rather than Iroquoian women specifically - saying they had "learnt the practice of procuring abortion by the use of some vegetable; and that it extends to prevent conception for a considerable time after" - though Mann points out that Jefferson isn't quite right here in regards to Iroquoian medicine. The herbs used for abortion were not the same as those used for contraception. In the 1800s, Dr. Peter Wilson (Cayuga) discussed the used of abortifacient herbal medicine, as did Arthur Parker (Seneca) in the 1910s-1920s.
After the American Revolution, the Haudenosaunee (the Iroquois Confederacy) broke up into three parts: one in New York, one in Ontario, and one in Ohio. In 1799, a new religious movement arose among the Iroquois still living in New York. The Gaiwi:yo - more commonly known as the Code of Handsome Lake, after its founder, or more generically as "The Longhouse Religion" - was a syncretic movement that blended traditional Iroquoian beliefs with Christian beliefs. Gaiwi:yo traditions became increasingly anti-abortion over time. Initially, it was only a mild encouragement for women to embrace their fertility and avoid medicines that would diminish their ability to have children (so note that this also includes contraceptives as well). By the 1990s, however, Gaiwi:yo preachers were siding with conservative Christianity and taking a strong stance against abortion.
Among Southern Coast Salishan groups, the spiritual world can seem all-encompassing. Omens covered everything from hearing an owl hoot (very bad luck) to coming across a fallen hummingbird nest (good luck for gamblers), spirits roamed the world, and one was never truly alone nor free from supernatural forces.
Yet, oddly enough (or not depending how one wants to look at it), abortion wasn't associated with anything particularly spiritual/religious...although acquiring abortifacients would have been seen in a similar vein to acquiring potions and ingredients for magic. Our sources are rare when it comes to the historical practice of abortion among the peoples that comprise the Southern Coast Salish, but those that are present paint a picture in which abortion appears to have significantly more to do with socially saving face than it did with the supernatural.
Two sources that do mention abortion are "The Puyallup-Nisqually" by Marian Smith, and "Chehalis Area Traditions" compiled from the notes of Thelma Anderson by Jay Miller. As such, I'll be going over what little they say and what that means within the contexts of historical Southern Salishan societies.
Puyallup-Nisqually
Within Marian Smith's 1940 anthropological survey of Southern Sound Salishan peoples, the use of abortifacients is explicitly mentioned twice:
"[Previous section discussed herbal remedies and potions meant to reverse the effects of menopause so older women can conceive] Another herb medicine for which a large price was also paid and which was known to only a few women, was said to have been an effective abortive. It was purchased secretly by the families of unmarried girls who had become pregnant."
...
"If the child was born, it was treated as any other child. Disapproval was visited not upon the child but upon the mother whose choice of husbands was narrowed. The use of abortives after marriage was denied and the suggestion was received with some surprise. Physical means for bringing on a miscarriage were apparently not resorted to." pp. 180
Whether abortion had similar restrictions and ill-tidings that death brought is not said (i.e. nothing suggesting that it invited the same sort of ritual cleansing that secretly murdering someone would, the loss of a soul that death among close family members occasionally brought, or that it affected one's power).
I will mention the implications of who is seeking out abortion and why further down.
Chehalis
In an ethnography of Chehalis (both Upper Chehalis and Lower Chehalis) and Sahaptin speaking Taitnapam villages, there is exactly one mention of abortion, with it being cited as a preferable solution to avoid social scandal:
"An unmarried royal girl killed her baby because she could never raise a bastard. She should have aborted it to avoid scandal." pp. 120
Notice what prominently comes up in these (admittedly very sparse) Southern Salish examples of abortion/reasons to seek it out? The families of young unmarried women are trying to obtain abortifacients, if they don't want the scandal of having illegitimate children then they should abort it. It is births to women out of wedlock among the nobility, scandalous sexual behavior among unmarried men and women that is a serious social no-no. Now for young women, this is not in the sense that one would be subject to stoning, mutilation, and being disowned by their families...it's that they almost permanently lose status and prestige which then impacts their lives in major and minor ways.
Your marriage prospects sink, people talk smack behind your back and occasionally to your face, other sorts of insults both subtle (i.e. poorer quality gifts at potlatches) and blatant are going to come your way, and your family has to deal with the stain that your actions have now brought upon them in addition to yourself. Now this isn't to say that your status drops to being akin to individuals who were/are slaves, slave descended, or lowborn. With time, the impact to your status and prestige fades to a fair degree as you make your mark on the community.
I guess a modern rough equivalent that one could think of would be being a convicted felon trying to just live life in the United States. Somebody who messed up and regardless of how they conduct themselves, face social stigma and ostracism.
Sources Used:
"The Puyallup-Nisqually", by Marian Smith
"Evergreen Ethnographies", by Jay Miller.