Organizers are angry the MLA for Highwood crossed the floor earlier this month to join the governing Progressive Conservatives.
Smith and eight members of her caucus made the move a week before Christmas.
Two of her constituents are behind the petition, which is also being made available online.
Amanda Achtman said the hope is to convince Smith to resign her seat and run in a byelection.
She said 40 volunteers were out in High River and Okotoks over the weekend knocking on doors to gather signatures.
"Her legacy now will be that she betrayed Albertans. She alienated friends, she took bad advice and she has departed from all of the principles for which she once stood. It's shocking. It's saddening."
Achtman said the physical petition had about 450 signatures by Monday morning while the online version had roughly 7,500.
Organizers are hoping to get 6,000 people in Smith’s riding to sign the petition, which would represent about 20 per cent of the electorate — a proportion Smith said should be the threshold to trigger a byelection in recall legislation she proposed as head of the Wildrose Party.
Anderson's 2nd floor crossing
In the last few months 11 Wildrose MLAs have crossed the floor, sparking some outrage in those ridings.
In Airdrie, voters have seen it before. Newly-minted PC MLA Rob Anderson has now switched parties twice, having originally crossed to the Wildrose Party from the Tories almost five years ago.
Some members of Airdrie city council are calling for a byelection, but Mayor Peter Brown said he likes having the local MLA in government.
"Selfishly as a mayor, I'm more than happy to see our MLA back with the government, negotiating and trying to secure some of the badly needed infrastructure,” he said.