Lengthy queues were building again on Saturday, following a day of bumper-to-bumper traffic on Friday.
It is one of the busiest periods for foreign travel from the UK as most schools in England and Wales have broken up for summer.
The Foreign Secretary said the delays and queues were “unacceptable”, blaming a lack of staffing by the French at the border.
But a French politician blamed Brexit for the chaos.
Pierre-Henri Dumont, Republican MP for Calais, said the problems at the Kent port would reoccur, telling BBC News: “This is an aftermath of Brexit. We have to run more checks than before.”
Mr Dumont also said the Port of Dover is “too small” and there are too few kiosks due to lack of space.
His comments linking delays to Brexit were backed up by the Port of Dover chief executive Doug Bannister, who said it must be recognised that there will be “increased transaction times” at the border due to extra checks needed.
Mr Bannister told the BBC Radio Four Today programme: “We are operating in a post-Brexit environment which does mean that passports need to be checked, they need to be stamped and indeed the capable people that do man the booths, police aux frontieres, they’re doing their job that they need to do now.”
He said the port had “created more border capacity so that the overall throughput can be maintained” and that while their modelling had shown some “very peak busy days during the summer season” are expected, “for the most part we should be able to cope with the traffic”.
His comments come after he said on Friday that being “let down” by poor resourcing at the French border was “immensely frustrating”.
Some 10,000 cars are expected to be processed going out of the port on Saturday, up on Friday’s figure of 8,500.
A “critical incident” was declared by the port due to the queues, with tourists urged to consider staying away, and warnings that Saturday may be just as bad.
Local MP Natalie Elphicke said “long, long delays” are expected again and insisted the French authorities “should apologise to Dover residents and holidaymakers for the unnecessary holiday chaos at the start of the Summer getaway”.
She also called for an end to “this sticking plaster approach – to invest in the roads, lorry parking & port facilities to support the @Port_of_Dover, Kent and Dover to grow & thrive”.
In a statement, Tory leadership hopeful Ms Truss said: “This awful situation should have been entirely avoidable and is unacceptable.
“We need action from France to build up capacity at the border to limit any further disruption for British tourists and to ensure this appalling situation is avoided in future.
“We will be working with the French authorities to find a solution.”
A French official said an “unforeseeable technical incident” in the Channel Tunnel meant that the staffing of border control booths had to be pushed back from 8.30am to 9.45am on Friday morning.
But Eurotunnel’s Director of Public Affairs John Keefe told the BBC that the Tunnel incident had “absolutely nothing to do with problems at the Port of Dover”.
Passengers embarking on cross-Channel sailings from Dover must pass through French border checks before they can board a ferry.
The port said in a statement that it had increased the number of border control booths by 50%.
There's been severe delays today at Dover & so I'm working closely with my opposite number @Cbeaune to address the issues that caused tailbacks.
I welcome his commitment that both Britain & France will work closely to minimise further disruption so people can get away quickly.
— Rt Hon Grant Shapps MP (@grantshapps) July 22, 2022
It went on: “Regrettably, the PAF (police aux frontieres) resource has been insufficient and has fallen far short of what is required to ensure a smooth first weekend of the peak summer getaway period.”
Jack Cousens, head of roads policy for the AA, said: “As the schools closed their doors fully yesterday, Saturday could prove busier still this weekend.
“Drivers should continue to expect disruption and delays on major holiday routes to the south-west, eastern coast and ports of Dover and Folkestone.
— French Embassy UK🇫🇷🇪🇺 (@FranceintheUK) July 22, 2022
“While many have decided to go at the start of the summer holidays, between now and the beginning of September when schools return, each Friday and Saturday will be busy on our roads.
“This is because these are the main switchover days for holiday lets.”
Ferry operator P&O Ferries told passengers to allow at least five hours to clear the approach roads and security checks.
The ferry company said on Friday evening it expects Saturday to be “just as busy”.
Grant Shapps said he was “working closely” with the French transport minister.
The Transport Secretary tweeted: “There’s been severe delays today at Dover & so I’m working closely with my opposite number Clement Beaune to address the issues that caused tailbacks.
“I welcome his commitment that both Britain & France will work closely to minimise further disruption so people can get away quickly.”
Labour’s Nick Thomas-Symonds accused the Government of being “absent”, amid another day of gridlocked roads, telling Times Radio that those in charge need to be “taking a grip of this situation”.
On Friday evening, the French Embassy in the UK said French border checks in Dover were “operating in full capacity”, adding that the French authorities were cooperating closely with their British counterparts and all stakeholders responsible for the traffic in the Port of Dover “to enable travellers to make their crossings under the best possible conditions”.