Not possible at all, unless we've got a PoD in late 1901/early 1902 that has her parents migrate to the U.S. - which in turn would result in a very different Alisa Rozenbaum, not the one we all know and hate.
She was a Russian woman with absolutely zero social skills and a total unwillingness to compromise. Those are four strikes against being a successful 20th C American politician.
Or indeed, a politician near any century.
But Ayn Rand does have some views that are inline with the American political scene - a staunch commitment to the use of Social Security disability, for one...
Her other views knock her out of the running
Easily enough done; the US goes Communist, and the remaining NATO countries seek to install someone with unimpeachable anti-Communist credentials to head up the USA remnant in the US Virgin Islands. Ayn Rand is routinely "elected" President of the USA by margins exceeding the number of eligible voters.
Ayn Rand would refuse to head a government in the "Virgin" Islands. A superior mind with the the will to achieve and dominate would never cling to the archaic moralistic trope of virginity.
Bahahaha!
I once had a tl idea in which after surviving the war, Quintin Roosevelt the war hero went to Hollywood during his longer lived father's second stay in the White House. He meets and marries Ayn Rand while there, enters politics as a war hero/showbiz star/Roosevelt Republican, then agrees to form a cross party ticket as VP candidate with cousin Franklin. When FDR croaks QR becomes POTUS with Ayn Rand as First Lady...
It's a little bizarre, but I got her into the White House. What do I win?