Members expressed frustration over Mr Sunak's victory despite being rejected by members two months ago.
While some felt "delighted" by the news, others said the move has left them "fuming" that party members were not able to vote for Mr Sunak to take over from Liz Truss.
Lyn Bond, a 60-year-old retired nurse who has voted Conservative since she was 18, sent an email to cancel her membership after it was confirmed the MP for Richmond will lead the party.
"The whole thing is rather sad because, for a party that had such a wonderful win in 2019, it has been destroyed from within," Ms Bond, from Bishop Auckland, said.
"I can’t bring myself after 40 years to support them anymore, I don’t trust them.
"I feel awash on a boat in the ocean not knowing where to go, what to do.
"I’ve never doubted what I voted for, ever, until today."
Ms Bond described Mr Sunak’s rise to prime minister as "insidious", explaining that party members feel like they "don’t know what’s going on behind the scenes."
Samuel Jukes, a retail worker from Birmingham who joined the Tories in 2019 and was supporting Boris Johnson in the contest, said a general election should now be called as Mr Sunak "has no mandate".
"I’m fuming right now, we never voted for Rishi Sunak," the 33-year-old said.
"I’m considering leaving, not decided yet but I see a lot of members are cancelling their memberships and I might be the same… Rishi has no mandate.
"Right now I’m ashamed to be a Conservative member."
But Tom Herman, from Hoveton, who is deputy chairman of North Norfolk Conservatives, said Mr Sunak’s road to becoming Conservative leader has been as "equally democratic" as a vote among party members.
Mr Herman said he is "delighted" that the leader has been selected so quickly, adding that Mr Sunak is "honest".
On party members not getting a vote, Mr Herman said North Norfolk Conservative MP Duncan Baker, who backed Mr Sunak, "consults very widely in the constituency, not just among Conservative Party members, and listens to those views and makes an informed decision on who to back for leader".
"That to me is equally democratic," Mr Herman said.
Roy Aldcroft, a Conservative councillor at Shropshire Council, said he was "pleased" about Mr Sunak’s confirmation as Tory party leader even though members will be "quite upset" that they were not able to vote him in.
"I’m very pleased it has been sorted out, that we will now have a clear path to sort out the many problems that we will be facing this winter," he said.
Mr Sunak will be formally appointed to the role in a handover of power overseen by the King on Tuesday.