Labour councillor Zafar Islam appeared to question the safety of the Covid jab in a series of tweets, saying people were concerned over the speed at which it had been developed.
He also raised concerns about "post-market surveillance arrangements and call-back procedures".
Lord Austin has criticised the Brierley Hill councillor, saying it was "very disappointing" to see him "undermine confidence in the vaccination programme".
However, Councillor Islam said he had recently received the first dose of the vaccine himself and had been encouraging people "vigorously" to take up the offer of the jab.
He said: "We are all concerned about the proper uptake of the vaccine because everyone's safety depends on all of us getting vaccinated.
"At the moment most of us are only having one dose, the other one is supposed to be in three weeks but unfortunately the period has been extended to 12 weeks.
"That in itself is making people doubt... if the manufacturers are saying it's three weeks then why are the government saying it is 12 weeks?
"But that's where we are, and I've had my first dose myself and my wife has had her first dose herself. We are very conscious that each one of us can contribute to safety.
"Whoever talks to us, we encourage people vigorously to take up the offer."
On December 31 official guidance was changed to say all second vaccine doses should be given after 12 weeks instead of three weeks to maximise the number of people protected.
The move has been supported by studies into the AstraZeneca and Pfizer jabs.
Councillor Islam said he encouraged all communities to take the vaccine, and added that there were "no religious issues as such with the vaccine in terms of the Islamic faith".
"The learned scholars have made it clear that there is no religious prohibition or dilemma in terms of what it contains or how it is made," he said.
"The bottom line is we all need to be safe and at the moment the only thing in town is the vaccine.
"It is really time to say, let's accentuate the positives of the vaccine and not give the negatives prominence.
"We ought to be saying to people who have taken the vaccine: you are leading by example.
"There's a huge task to be finished with 60 million people to be vaccinated. If we keep talking about the negatives it will be slowed down."
Marco Longhi, the Conservative MP for Dudley North, described the councillor's tweets as "dangerous rhetoric" that could "potentially put people’s lives at risk".
He also questioned separate tweets by Councillor Islam which called for Covid vaccines to be rolled out on the Gaza Strip.
Mr Longhi said: "One minute he is attacking the vaccine and the next he is pushing for the vaccine in the Gaza.
"After promoting critical material about the life saving vaccines, he has now done a U-turn for political purposes. This is politics of the lowest level, playing on people's fears and trying to make political gain.
"I believe this is real hypocrisy and his Labour colleagues must distance themselves from such material."