According to the online Cambridge Dictionary the idiom be on (someone) is “used when saying who is responsible for something.” It mainly occurs in informal USAmerican speech.
Take your best guess at what the number and costs should be – add 5% to be safe – and tell them that that's all they get for the year and it's on them to make it work. — Howard Tullman, The Perspiration Principles XI: You Get What You Work For, Not What You Wish For, 2015, 50.
After being selected by the New York Knicks with the 3rd overall pick in this year’s NBA draft, R.J. Barrett has a lot of new fans and critics alike. The rookie is going to receive a lot of attention in his first year in the league, and it’s on him to step up. — Zachary Bachar, “New York Nicks: Three Realistic Rookie Season Goals for R.J. Barrett,” Empire Writes Back, 10 Aug. 2019.
It's on you to create a brand people can care about. — John Michael Morgan, Brand Against the Machine, 2011.
Although the Cambridge Dictionary suggests a primarily American usage, the earliest examples I could find were from Australia. This likely has more to do with the nature of the search and the informality of the idiom — coupled with an inferior search engine at the Library of Congress newspaper archives — than a suggestion of provenance.
… I am going to say it is on her to prove that she was insane beyond reasonable doubt. — “Insanity and Divorce,” The Register (Adelaide, SA), 18 Apr. 1923.
It was a great reversal for Port, and although they can stand the defeat on the competition table, it is on them to continue in the winning way. — Illawarra Mercury (Wollongong, NSW), 12 July 1935.
In more formal language — and before the rise of the simpler idiom — what is on/upon a person or group was a limited number of nouns, such as onus, burden, obligation, duty, responsibility, or the adjective incumbent:
… and further, where such grave charges are made, the onus is not on those who have to answer them, but the onus is on those who make them, to support them, and until they are supported by evidence, so cogent as to call for an answer, individuals are not put in the peril of making an answer at all. — M. Hobart Seymour, The Talbot case : an authoritative and succinct account from 1839, London, 1851.
Although we do not wiſh to take a part in the diſpute reſpecting colonel Burr, farther than to repel inſinuations which we know to be groundleſs, we cannot refrain from ſaying generally, that much more was incumbent upon thoſse who made ſo heavy a charge againſt colonel Burr, and pledged themſselves to prove it by testimony, than merely to refer to gentlemen who it is pretended are acquainted with the facts.
— Republican, & Petersburg Advertiser (VA), 21 Oct. 1802.