The Commons Public Accounts Committee praised the "world-beating" effort to get the jab to the most vulnerable, but warned it was essential not to lose momentum.
The Government has said it is on course to meet its target of offering a vaccine to the estimated 15 million people in its top four priority groups - including frontline health and care workers and the over-70s - by Monday.
However the committee said there was still "much to be done" to hit its next target of getting the jab to the 17.7 million in the next five priority groups - including all over-50s - by the end of April.
"We are concerned by departments' lack of planning for the next phase of the programme and in learning the lessons from what has already been done that will be so vital to the programme's success," it said.
Despite the confidence of the Department for Business, Energy and Industrial Strategy (Beis) that the UK has access to more than enough doses, the committee said there were "concerns" over the supply chain.
It said Ministers needed to ensure plans were in place to respond to potential future developments such as the need for an annual vaccination programme or the discovery of new variants of the virus.
The committee said the Government would continue to face "significant challenges" in ensuring it gets the jab to "the right people at the right time", particularly given the different handling requirements of the Pfizer and AstraZeneca vaccines.
There was a risk that its plans for the programme "will not meet public expectations", the committee said as it stressed the need for clear messaging.
"Government has at times struggled to communicate clearly to the public about what they can expect from the vaccine programme, otherwise it risks confusion about who will be able to access the vaccine, how and when," it said.
"With misinformation about vaccines being circulated on various digital platforms, clear communication from Government is particularly important to maintain public confidence and take-up."
One of the committee members, Labour MP for Sheffield Hallam Olivia Blake, said people "on the ground" in her constituency were "very happy" with how the programme is going.
She said: "At the time that we were taking evidence there was a bit of confusion, amongst health and the public about messaging, about who would get the vaccine when and the 12-week wait and all these different issues.
"We were quite clear in our recommendations that communication is absolutely key for this to continue to have public support, and I think they have to step up and continue to step up to make sure that they're able to communicate any changes to the programme going forwards as well."
The committee said there was a "strong case" for looking again at which groups should be prioritised after the most vulnerable have been vaccinated - especially frontline key workers who are more exposed to community transmission of the virus.
It also said the Government "could have been more transparent" about how decisions relating to the programme had been made.
It was unclear why the original head of the Vaccine Taskforce, Kate Bingham - who was appointed by Boris Johnson - was chosen while almost a fifth of the taskforce members had recorded at least one conflict of interest, although most were "minor".
Committee chair Meg Hillier said: "We recognise the huge efforts made by all those involved in developing, procuring, testing and delivering the vaccines.
"The UK has been at the forefront of the global effort to find and deliver a vaccine but now is not the time to rest on our laurels - Covid-19 remains a significant threat to our nation's health and economy.
"Vaccination is a key component in meeting that threat and the committee calls on the Government to build on this strong start, including building on work to boost the UK manufacturing base."
The Department for Health and Social Care has been contacted for a comment.