It is not true, though it's widely claimed, even in academic sources.
Robert J. Priest, an anthropologist and a missionary himself, sought out the origin of the term in the 1990’s, something that no one else had apparently done. The idea of his paper, "Missionary Positions: Christian, Modernist, Postmodern" (available on academia.edu), is that "missionary position" was used repeatedly to criticize missionaries by anthropologists after a major turn in anthropology in the 1960's and 1970's (for more on this turn, I strongly recommend Sherry Ortner's "Theory in Anthropology Since the Sixties" and its follow up "Dark Anthropology and Its Others: Theory in Anthropology since the Eighties").
While it's often attributed to missionaries elsewhere, the kernel of the idea arose from missionary encounter in the South Pacific, rather than Africa or the Americas. However, even this idea is misleading: the term arises from a misreading or misremembering of the famous “Kinsey Report”, in which Alfred Kinsey himself misremembers Malinowski’s famous research in the South Pacific. That is to say, it’s the result of a 1960’s and 70's popularization of a 1948 misunderstanding of research of published in 1929 based on ethnography done in the 1910’s.
Let's start with Malinowski, as Priest finds that he's the only actual primary source, ethnographic or otherwise, ever associated with the use of the term the "missionary position" and actual missionaries. Malinowski references two separate ideas in his Sexual Life of Savages in North-Western Melanesia that Kinsey later combined and accidentally elaborated on.
After going through a description of three sexual positions which are used on the Trobriand Islands, Malinowski goes on to say:
No other positions are used. Above all, the natives despise the European position and consider it unpractical and improper. The natives, of course, know it, because white men frequently cohabit with native women, some even being married to them. But, as they say: “The man overlies heavily the woman: he presses her heavily downwards, she cannot responds (ibilamapu)."
Altogether the natives are certain that white men do not know how to carry out intercourse effectively.
Malinowski goes on to make it clear that the natives are mocking sailors, as well as "white traders, planters, [and] officials", rather than missionaries. The local knowledge of the position comes through gossip after these white sailors and traders having sex with native women, rather than more direct observation or prudish missionary instruction. While the idea of the missionary position in the popular and academic imagination is that it's supposed to have been an attempt by missionaries to "civilize" savages into controlling their more base and sinful sexual urges, that's not how the Trobriand Islanders saw the position at the time. They, instead, saw the European style of having sex as deficient in every way. It's something to be mocked, not emulated. And it's not just the position, but the whole way of having sex. Malinowski mentions, for example, “to the native idea, the white man achieves orgasm far too quickly," among other criticisms. It is not seen as the mark of civilization or refinement or a new colonial order, and it's not taught by missionaries; it's just seen as wrong and clumsy, and taught by men having sex with local women.
The second set of incidents from Malinowski that get caught up in the “missionary sex” idea isn't even about sex directly. Later in the volume, Malinowski discusses courtship rituals in Melanesia. After discussing the intense privacy of Trobriands during the actual acts of physical intimacy (as opposed to the very open talk of sex), he discusses how public displays of affection common in the contemporary European parks (by our standards, surely very demure displays of affection indeed) would never occur in Melanesia because they would be too scandalous.
Holding hands, leaning against each other, embracing [...] are not permitted to lovers in public. I observed once or twice that Yobukwa'u and his betrothed used to lie together on a mat in broad daylight, decorously, but unmistakably leaning against each other and holding hands, in a manner which we would find perfectly natural in a pair of lovers soon to be married. But when I mentioned this in discussing the whole subject with natives, I was total at once that it was a new fashion and not correct according to old custom. Tokolibeda, once a famous Don Juan, now a peppery old conservative and a stickler for proprieties, insisted that this was misinari is bubunela, "missionary fashion", one of those novel immoralities introduced by Christianity. He spoke with as much feeling and righteous indignation as the Late Rev. C. M. Hyde of Honolulu might have used against heathen pruriency.
One gets the sense, though, that missionaries weren't actually teaching this as ideal behavior. Likely, this was picked up by observing the various missionaries, colonial officials, sailors, and other Europeans. These are acts considered private in Melanesian culture but performed publicly by Europeans. Young Melanesians seem to have begun performing them publicly, in the European fashion, in this period.
So, on the one hand, we have face-to-face sex and other sexual styles of European sailors that are mocked for being ineffective and on other hand we have public displays of affection like cuddling that are condemned by old conservatives for being prurient. So in Malinowski we have the "European position" that missionaries likely don't engage in, and the "missionary fashion" that isn't sex at all.
Almost all contemporary sex research has a genealogy that goes through Kinsey, and he misremembers and combines these two points from Malinowski. No lexicographer, despite extensive searching, dates the term before Kinsey. Kinsey doesn't even call the position the "missionary position" for the most part; he primarily calls it the "English-American" position. In his 1948 book, Sexual Behavior in the Human Male, in the section called "Positions in Intercourse", he writes:
Universally, at all social levels in our Anglo-American culture, the opinion is held that there is one coital position which is biologically natural, and that all others are man-devised variants which become pervasions when regularly engaged in. [...]
Most persons will be surprised to learn that positions in intercourse are as much a product of human cultures as languages and clothing, and that the common English-American position is rare in some other cultures. Among several thousand portrayals of human coitus in the art left by ancient civilizations, there is hardly a single portrayal of the English-American position. It will be recalled that Malinowski (1929) records the nearly universal use of a totally different position among the Trobrianders in the Southwestern Pacific; and that he notes that the caricatures of the English-American position are performed around the communal campfires, to the great amusement of natives who refer to the position as the "missionary position".
Priestly notes that the public mockery ritually occurs under the full moon, rather than a campfire, in Malinowski's telling and it is not here, but elsewhere, that Malinowski notes the specific mockery of the sailors' "European position". So rather than just combining two parts of Malinowski's work, he's actually combining three. Kinsey speculatively continues:
The origin of our present custom is involved in early and later Church history, and needs clarification before it can be presented with any authority; but certain it is that there was a time in the history of the Christian Church when the utilization of any other except the present-day position was made a matter for confession.
I am unaware of any research that indicates that such a thing ever happened (perhaps my medieval authority, /u/sunagainstgold knows more, but I sort of doubt there's much more to know), but nonetheless it begins to be treated with authority by later authors, many of whom are writing only half-remembering their Kinsey who was only half-remembering his Malinowski.
(continued below)