The petition, which is on BCP Council's website, had accumulated 181 signatures by Wednesday evening. It runs until 30 June.
Morebus operates the 737 service for the route.
Richard Wade, general manager for the group, said it "would be keen to operate a trial service should funding become available for such a route".
Bournemouth, Christchurch and Poole (BCP) Council said the authority would "welcome enhanced bus services to and from the airport".
"Bus services are not operated by the council – they are provided by commercial operators and whilst the council can help subsidise some routes we are limited in what we can do," a spokesperson said.
He said that it should not be complicated to arrange more buses, as there is a route already in place.
A Bournemouth Airport spokesperson said: "We are aware of concerns around public transport access to the airport and are in active discussions with local partners, including BCP Council, to explore improved connectivity.
"This includes better public transport links and wider opportunities for improved pedestrian and cycle access as part of future development plans."
