Long Melford Heritage Centre, Sudbury Town Council and the Daws Hall Trust are all hoping for a cash boost from the Tesco Bags of Help initiative.
Grants of £4,000, £2,000 and £1,000 – raised from the 5p bag levy – will be awarded to community projects according the number of votes each scheme receives.
The heritage centre is one of three groups to have been shortlisted for the scheme.
Funding would go towards its First World War project, telling the story of local men and women, both those who returned, and those who did not.
Sudbury Town Council is also on the shortlist. It hopes grant funding will allow for the repair of The Croft boating pond.
Assistant town clerk Jodie Budd said: “The base of the boating pond is currently unsuitable for its purpose and desperately needs repairing.
“Unfortunately, as with all councils, funding is limited so we are hoping this scheme will kick start it.”
Daws Hall Trust in Bures is looking to use grant money to help the centre run open days and events for the community.
Voting is open in Sudbury’s Tesco store throughout May and June.