The influential Public Accounts Committee said if upgraded controls were not in place, British ports would see huge queues and businesses would face massive disruption.
HMRC maintains that the CDS programme is on track and is well governed, but it admits that major risks remain, which means that CDS might not be fully operating by the planned date of January 2019.
Food could be left to rot in trucks if a Brexit deal is not struck, it's been claimed.
"HMRC is under considerable pressure to deliver the new Customs Declaration Service in time, but it does not yet have funding to increase the capacity of CDS to deal with the consequences of Brexit - nor to develop contingency options".
"This report makes for stark reading for businesses and consumers who are dependent on moving goods quickly between the United Kingdom and the rest of Europe".
The committee said, "Under current plans, the United Kingdom is set to leave the European single market and the customs union in March 2019".
The move to the updated CDS system was planned before the United Kingdom voted to leave the European Union, and following the referendum result, its completion date was brought forward to January 2019.
The House of Commons Public Accounts Committee said the number of customs declarations which HM Revenue and Customs must process each year could increase nearly five-fold after the UK's departure from the European Union - from 55 million to 255 million.
It said: "This is a programme of national importance that could have a huge reputational impact for the United Kingdom if it is not delivered successfully".
Negotiations to replace the UK's customs union and single market membership have not yet started, with agreement not yet reached in the initial phase of talks. As part of its plans to broaden engagement with traders, it should also promote the benefits of obtaining trusted trader status and aim to increase the number of registered traders, the PAC said.
But MPs have warned that the current systems in place to handle this traffic are incapable of managing such large amounts of work - and urgently need to be bolstered. It argues that HMRC must prioritise CDS to ensure a successful delivery.
"The UK can not afford for this to go wrong".
Director general, Robert Keen said: "We share the Committee's concern that HMRC does not yet have the necessary funding and resources to produce the infrastructure that will be required to facilitate Customs processes post-Brexit".
A government spokesman said the CDS would be able to cope with a "significant increase" in customs declarations at the UK's border.
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