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7 year oldKory Langhofer made the comments in a letter to congressional committees.
But a spokesperson for Mr Mueller said the "appropriate criminal process" had been followed.
Mr Mueller is investigating allegations of collusion between Mr Trump's presidential campaign and Russia.
Mr Langhofer works for the Trump for America (TFA) group. It used the facilities, including email hosting, of a government agency, the General Services Administration (GSA), in the period between Donald Trump's election in November 2016 and inauguration in January.
In his letter, Mr Langhofer, says GSA staff "unlawfully produced TFA's private materials, including privileged communications, to the special counsel's office".
The GSA, he complains, "did not own or control the records in question" and the constitutional rights of transition officials were violated.
Mr Langhofer goes on to ask Congress to act to protect future presidential transitions from having "private records misappropriated by government agencies, particularly in the context of sensitive investigations intersecting with political motives".
The emails obtained reportedly involve 13 Trump transition officials, including former national security adviser Michael Flynn who pleaded guilty to making false statements to the FBI earlier this month.
A spokesperson for Mr Mueller said they had done nothing wrong.
"When we have obtained emails in the course of our ongoing criminal investigation, we have secured either the account owner's consent or appropriate criminal process," Peter Carr said.
Of course Mueller obtained emails from a third party. Prosecutors in most white collar criminal investigations do that. It’s not “inappropriate” or even unusual. Anyone who claims otherwise has no idea what they’re talking about. https://t.co/1JKNobvKJd
— Renato Mariotti (@renato_mariotti) December 16, 2017
GSA Deputy Counsel Lenny Loewentritt has denied another of Mr Langhofer's accusations, that the GSA assured that requests for Trump transition records would go through the group's lawyers.
Democratic Representative Eric Swalwell tweeted that the accusations were "another attempt to discredit Mueller as his #TrumpRussia probe tightens".
US intelligence agencies believe Moscow tried to tip the presidential election in favour of Mr Trump - a charge denied by both Russia and the US president.
Mr Trump has called Mr Mueller's investigation a "witch hunt" while other Republicans accuse it of bias.
Michael Flynn became the most senior Trump official to be charged as part of the inquiry after admitting making false statements to the FBI about meetings with Russia's ambassador.
Another ex-aide, George Papadopoulos, has also pleaded guilty to making false statements to FBI agents.
In October, Mr Trump's former campaign manager, Paul Manafort, and his business associate Rick Gates were accused of conspiring to defraud the US in dealings with Ukraine. Both deny the charges, which centre on relations with a former Ukrainian president who was very close to Russia.
On Friday, the top Democrat on the House intelligence committee, Adam Schiff, said he feared Republican members wanted to shut down their own investigation.
President Trump's private lawyers are expected to meet Mr Mueller and members of his team next week to discuss the next phases of the inquiry, US media report.