Councillor Shaukat Ali, Labour's health chief, made the suggestion last month to take the vulnerable to get their vital jab.
But council leader Patrick Harley said the last thing people needed was every "Tom, Dick or Harry" offering to drive them.
Now Councillor Ali – after speaking with health experts – said the move was allowed if people meet certain safety rules.
He said: "At long last public health advice has emerged with what I had anticipated, a common sense approach.
"My suggestion was not taken in the spirit with which it was given and the insulting and derogatory comments from the Conservative leader of the council were uncalled for.
"Volunteers have gone over and beyond the call of duty ever since the first lockdown helping vulnerable members of our community. To call them 'Tom, Dick, Harry' in a derogatory manner is unacceptable. The leader has not only shown his ignorance but how out of touch he is with ordinary people.
"I am pleased that my persistence has resulted in support being made available to those who need it most."
The advice – available to volunteers, family members and taxi drivers – state there should be good ventilation in the car by opening the windows and the passengers sit behind the driver on the left.
Seatbelt buckles and handles should be wiped down after use with a chlorine based detergent, masks must be worn throughout the journey and people should use hand sanitiser – or frequently wash their hands.
Councillor Harley, leader of Dudley Council, said the suggestion was still very risky – with the most vulnerable still at risk from the pandemic.
He added: "What Councillor Ali was suggesting at the time was actually illegal so it comes as no surprise to anyone that I wouldn't support his suggestion.
"There are and have been for a long time now GP surgeries that have practice nurses that will carry out home visits.
"Home visits by trained medical staff without the risk provided by Councillor Ali's suggestion where a vulnerable adult could easily risk being infected by someone – no matter how helpful because the vehicle or themselves hadn't taken all of the necessary precautions.
"The vaccination roll out in Dudley has been a tremendous success with 100 per cent of 75-79 and the 80-plus age groups receiving the first vaccine.
"We also have over 80 per cent of 70 to 75-year-olds enjoying the first jab. Therefore, what he is suggesting has very little relevance as the amazing work of our NHS and public health teams continues at a blistering pace."