The vacancies are in the Stratton ward of the town council.
The deadline for electors to request an election is April 20 - if at least ten electors have contacted the Proper Officer at Cornwall Council by then requesting an election, one will be held.
If this does not occur, vacancies will be filled by co-option.
In a statement, Bude-Stratton Town Council said: "Due to the resignations of Martin Dorey and Paul Tilzey, two vacancies have arisen in the Office of Councillor for the Stratton Ward of Bude-Stratton Town Council.
"If by 20 April, the request for an election has been received from at least 10 electors by the Proper Officer in Truro, an election will be held to fill both vacancies. Otherwise the posts will be filled by co-option. Bude-Stratton Town Council would like to thank both Martin and Paul for their time on the Council and wish them all the best in their future endeavours.
"Full details of the process can be found in the Notice of Vacancy here https://bit.ly/3G0D7qS"