Is it possible to reach Lavenham by public transportation? I have no intention to drive.
Yes. How would depend on where you are coming from.
There is a bus service that runs from Colchester to Bury St Edmunds via Sudbury and Lavenham. Colchester, Bury and Sudbury all have train stations. From London, you would take a train to Colchester, at which point you could take the bus to Lavenham, or you could catch the Sudbury train, and join the bus there.
Bus schedule at www.chamberscoaches.co.uk/html/753.html
Train schedule and fares at www.nationalrail.co.uk