It comes after walk-in centres opened for all over-40s in Edinburgh on Wednesday.
Labour Lothian MSP Sarah Boyack said these clinics do not go far enough, and that “walk-in clinics for over-18s are now a matter of urgency.”
“We have been asking for walk in clinics for hotspot areas and we need to see a faster roll out of the vaccine in Lothian,” she said.
Ms Boyack asked the First Minister on Tuesday when clinics would be opened to over-18s, but said she did not receive an answer.
The First Minister said NHS Lothian “have plans in preparation for walk-in vaccination clinics.”
Ms Boyack said on Wednesday: “Although drop-in vaccination clinics are being rolled-out for people aged over-40, the First Minister did not answer my specific question about whether the Scottish Government has plans to set up walk-in clinics in Lothian for over-18s.
“We urgently need clarity on when can expect the rollout.”
Liberal Democrat Alex-Cole Hamilton, who previously called for walk-in centres, said that while he welcomed the clinics for over-40s the measure must be extended to lower age groups.
"With the case rate in Edinburgh doubling in the last week, we don't just need walk in centres for those who are over-40, we need them for everyone,” he said.
"We know that younger people tend to move more frequently and blue letters are less likely to reach them. Making walk-in centres available for everyone would help in solving this issue.
"Thousands of my constituents are desperate to get vaccinated, they just haven't been given the opportunity.
"The Government should also learn from other parts of the UK and roll out door to door surge testing in the areas with the highest caseloads."
The seven day rate of Covid cases in Edinburgh has risen to 181 per 100,000 population, compared to 135 this time last week. The rate across Scotland as a whole is 99.
The Scottish Government and NHS Lothian were contacted for comment.
Dona Milne, Director of Public Health and Health Policy, NHS Lothian, said: "Drop-in clinics for over-18s were only introduced in other areas of the country as part of a response to a worrying surge in cases, when the number of linked cases had risen rapidly in a particular area.
"In Edinburgh, our cases are scattered across the city. We are working really closely with partners to monitor Covid-19 patterns and trends in Lothian and there are robust plans in place if any similar activity was required.
“We are working through the vaccination priority groups in line with national JCVI guidance and we are vaccinating as many people within those “cohorts” as quickly as the vaccine supply will allow. In Lothian and the rest of Scotland, we are already preparing appointments for those aged 18-29 and asking that age group to register with us urgently so that we can do this quickly.
“In addition to vaccination, additional mobile testing units and Lateral Flow Device Kits can be picked up at all community testing sites. We would encourage people to get tested if they think they have been at risk.”