Mr Baxter's bronze was the first ever Olympic snowsports medal won by a British skier or boarder but the 40-year-old's historic achievement 12 years ago at Salt Lake City in America was tarnished by controversy after he tested positive for banned substances.
The campaign comes after snowboarder Jenny Jones landed a bronze at the current Sochi Olympics in Russia, leading to statements from the head of Team GB that her medal is the first ever won by a British athlete on snow. The comments have revived annoyance in some areas of the grassroots snowsports community who believe Baxter should have his medal restored.
The online petition to overturn the decision has attracted nearly 2000 signatures in just two days.
Mr Baxter said yesterday that he was "overwhelmed" by the support and he had "not given up hope" of "justice being done".
He is taking his three young children to CairnGorm Mountain on Friday to ski the slopes there - and Baxter said he wanted the bronze medal for his children to realise he had not been a cheat.
Baxter's bronze medal was taken off him when he failed a post race drugs test after his slalom success in Utah.
It was established that he failed because he had bought an over-the- counter Vick's inhaler in America which included the banned substance levomethamphetamine.
British Vicks inhalers do not include levomethamphetamine and Baxter had not realised there would be a difference.
Levomethamphetamine has been shown to have zero performance-enhancing qualities.