It follows a decision by the Conservative Party leader to duck one-on-one interviews with the BBC's Andrew Neil and ITV's Julie Etchingham during the election campaign.
He has also turned down invitations to appear on ITV's Good Morning Britain and was accosted live on air on Wednesday by one of the programme's reporters.
"For five weeks, we had high hopes of interviewing the Prime Minister but despite being led on by his advisers that it was going to happen, we have learnt it is not going to," he said.
"We are a little dismayed on your behalf."
All the leaders of the six other major parties - Labour, SNP, Liberal Democrats, Green Party, Plaid Cymru and the Brexit Party - had taken part or agreed to an interview on the show.
According to the latest audience figures, 14.5 million listeners tune into Radio 2 every week, making it the nation's favourite station.
"Over the last two weeks, this programme will have interviewed six of the leaders of the main political parties in this country," Vine told listeners.
"Until yesterday afternoon we had every reason to think that today I would be sitting here opposite the Prime Minister Boris Johnson.
"Yesterday afternoon we were told by the Prime Minister's press officer he was pulling out and it wasn't going to happen.
"That means that millions of Radio 2 listeners are not going to have the chance to hear the PM being scrutinised about his policies, his character and his plans for the future on this programme."
Vine, who also has a television show on Channel 5, said his programme had tried to focus on stories "outside the Westminster bubble" during its election coverage.
The talk-and-music show host said he and his predecessor Jimmy Young had, over a 50-year period, interviewed every sitting prime minister during an election campaign since Margaret Thatcher - until two years ago.
"Two years ago, Theresa May declined to come on - it was no secret she was uncomfortable doing leader interviews," he added.
"Boris Johnson, on the other hand, has always been very media friendly, maybe even media hungry."
Jeremy Corbyn also faced criticism after LBC radio presenter Iain Dale said the Labour leader was avoiding call-in programmes - unlike Mr Johnson, who faced awkward questions about how many children he has during an hour-long LBC phone-in.
Former Tory candidate Dale tweeted: "More stories today about the PM not doing an interview with Jeremy Vine.
"Little said about Jeremy Corbyn refusing to do a phone-in with either LBC or 5 Live.
"Corbyn hasn't done a live interview with any LBC presenter since he's been leader. Quite an insult to our 2.7m listeners."