"act of obtaining an interest in," 1970, from verbal phrase buy in "to purchase a commission or stock" (1826), from buy (v.) + in (adv.). To buy into "obtain an interest in by purchase" (as of stock shares) is recorded from 1680s.
Old English bycgan (past tense bohte) "get by paying for, acquire the possession of in exchange for something of like value; redeem, ransom; procure; get done," from Proto-Germanic *bugjan (source also of Old Saxon buggjan, Old Norse byggja, Gothic bugjan), which is of unknown origin and not found outside Germanic.
The surviving spelling is southwest England dialect; the word was generally pronounced in Old English and Middle English with a -dg- sound as "budge," or "bidge." Meaning "believe, accept as true" is attested by 1926. Related: Bought; buying. To buy time "prevent further deterioration but make no improvement" is attested from 1946.
a Middle English merger of Old English in (prep.) "in, into, upon, on, at, among; about, during;" and Old English inne (adv.) "within, inside," from Proto-Germanic *in (source also of Old Frisian, Dutch, German, Gothic in, Old Norse i), from PIE root *en "in." The simpler form took on both senses in Middle English.
Sense distinction between in and on is from later Middle English, and nuances in use of in and at still distinguish British and American English (in school/at school). Sometimes in Middle English shortened to i.
The noun sense of "influence, access (to power or authorities)," as in have an in with, is first recorded 1929 in American English. to be in for it "certain to meet with something unpleasant" is from 1690s. To be in with "on friendly terms with" is from 1670s. Ins and outs "intricacies, complications of an action or course" is from 1660s. In-and-out (n.) "copulation" is attested from 1610s.
<a href="https://www.etymonline.com/word/buy-in">Etymology of buy-in by etymonline</a>
Harper, D. (n.d.). Etymology of buy-in. Online Etymology Dictionary. Retrieved $(datetime), from https://www.etymonline.com/word/buy-in