The words “Murder” and “Go vegan” have been spray-painted on the door at Greendale Farm Shop in Woodbury Salterton, Devon.
Greendale is providing a service where people can name their own turkey and “help look after it for the next two months”.
The Vegan Society said it “does not support” any threats or abuse.
Farmer Mat Carter, 35, said he wanted to “offer people the chance to see where their food came from”.
He said the backlash to the service started “almost immediately” after it was shared on Facebook and Instagram on Sunday.
He said they had also received “very strange” phone calls.
One person reportedly called and asked the butcher: “How would you like it if I cut you up and put you on the counter?”
Another caller is said to have shouted obscenities, before saying: “You should be the ones being killed because your life is worth less than that of the innocent animals which you are murdering”.
Despite the backlash, Mr Carter told the BBC staff would not give in to any pressure.
He added that turkey sales “have gone through the roof” and the shop is “busier than it has ever been”.
The Vegan Society said it “does not support any threats of violence or abusive behaviour or language towards anyone”.
It encouraged “vegan activists” to share their message “peacefully and positively”.
“It’s important to note that veganism is based on kindness and compassion, with the vast majority of vegans campaigning this way,” it said.
“There are extremists in every movement but they are not representative of the movement as a whole.”