Stein's campaign announced Thursday that it had raised the $1.1 million necessary to fund a recount in Wisconsin and was on track to do the same in Pennsylvania and Michigan, having raised $2.7 million of a $4.5 million goal.
Her campaign has cited unspecified "anomalies" as grounds to mount a challenge in all three states.
The move comes amid stepped-up calls from some of Trump's left-wing opponents to challenge the results of the November 8 election, which was marked by warnings of foreign hacking and vote rigging.
"The unexpected results of the election and reported anomalies need to be investigated before the 2016 presidential election is certified," Stein said on her website. "We deserve elections we can trust."
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Although there is virtually no chance of overturning the result, the demands could reignite debate over the legitimacy of Trump's election, already fuelled by Democrat Hillary Clinton's lead in the popular vote which now stands at two million.
"Raising money to pay for the first round so quickly is a miraculous feat and a tribute to the power of grassroots organizing," Stein's website read.
The Midwestern state of Wisconsin was a key battleground in the November 8 election, helping propel the Republican Trump to a victory that stunned the nation.
Trump also won swing state Pennsylvania, and claimed a razor-thin victory in Michigan according to unofficial results released Wednesday.
Although she lost in the indirect electoral college tally which decides who wins the White House, Clinton's lead in the popular vote has risen as official results are finalised, narrowing the result in some swing states where she lost.
Clinton conceded on November 9 and her campaign is not supporting the recount calls.
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'Election integrity movement'
Stein's demands follow a bitter presidential campaign that included persistent charges of Russian hacking and fraud.
Trump also warned during his campaign that the election would be "rigged," and refused to say whether he would accept the results.
Any result change would not help Stein, who won just over 1.1 percent of Wisconsin's vote.
But she launched the recount effort after computer scientists reportedly told the Clinton campaign the election may have been rigged in Trump's favor.
The scientists, including University of Michigan's J. Alex Halderman, together with voting rights activist John Bonifaz and others, said Clinton's votes were 7 percent below expectations in counties that tallied votes with electronic machines.
"These recounts are part of an election integrity movement to attempt to shine a light on just how untrustworthy the US election system is," Stein's website said, adding that the attempt is "not intended to help Hillary Clinton."
Clinton lost by a razor-thin margin of around 27,000 votes in Wisconsin and 60,000 votes in Pennsylvania.
Unofficial results released by Michigan's secretary of state's office, put Trump ahead by a mere 10,704 votes.
"I think it's only natural and good for Americans to be reassured that our votes are counted," Stein told CNN Thursday, "especially after such a divisive and bitter election where 80 percent of Americans... basically said they were disgusted with this election."
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'Demographics, not hacking'
But one of the scientists she cites, Halderman, Wednesday sought to dampen mounting speculation that recounts would change the election's result.
"Were this year's deviations from pre-election polls the results of a cyberattack?" he wrote on Medium. "Probably not."
Rather, the polling was more likely "systematically wrong."
The only way to remove any lingering controversy, however, would be to study "paper ballots and voting equipment in critical states," he added.
The well-known statistician Nate Silver was also skeptical about any possible change in the results, tweeting that the disputed voting patterns "are well-explained by demographics -- not hacking."
Wisconsin swung Republican this month for the time since voters there helped elect Ronald Reagan in 1984.
It's unclear how much success Stein will have challenging the results in other states.
A recount filing costs $500,000 in Pennsylvania, due on November 28. The filing fee in Michigan is $600,000, due by November 30.
Stein's campaign anticipates eventually needing $6 million to $7 million to cover attorney fees and statewide recount observation costs
