Kent police said they were aware of the protest, which the Port of Dover described as a "public order incident". Officers were in attendance and engaging with protesters, they said.
Traffic in and out of the port on the A20 has been stopped, causing delays of up to 30 minutes and four miles of congestion, according to National Highways.
Videos and pictures shared online show a large number of masked people in black marching on the road in a large group as lorries in traffic next to police cars.
The number of people taking part in the protest is unclear. It is reported to be dozens by PA. One of those filming the event estimated he was marching alongside 800 to 1,200 people.
The far-right activist Tommy Robinson, whose real name is Stephen Yaxley-Lennon, reposted a video of the blockade, highlighting a livestream of the protest.
He encouraged more to join, saying "get there to support them".
Some have estimated that about 300 people have gathered, although this has not been confirmed by police.
The Port of Dover confirmed in a statement that the disruption was affecting "all routes" in and out, as well as around the town, and that it was in touch with the police.
It said: "The Port of Dover can confirm that there is an ongoing public order incident happening around the town of Dover. This situation is impacting our local community and our customers, so of course we are hoping for a swift resolution."
Nathan Ditchburn, who was on his way to work when he saw the group arrive at about 6.45am, told KentOnline: "I would say there are around 300 men, maybe more. Nobody will speak; they just handed me a piece of paper to look at. They have blocked off every entrance and exit of the Dover port area, and Jubilee Way is blocked as well. It will be causing huge delays for the ferries today."
The local parkrun group cancelled its Dover Waterfront run, telling runners the course was unsafe or blocked. A spokesperson told KentOnline: "I am afraid we are going to cancel this morning. We have mass disruption with some protesters."
Mike Tapp, the Labour MP for Dover and Deal, criticised what he described as "idiots coming to Dover to cause disruption". In a post on Facebook, he said: "Parkrun cancelled and port access disrupted. I am fed up with idiots coming to Dover to cause disruption. I bet if you asked any one of those 'protesters' they wouldn't know small boat crossings are down by 40% and we are working hard to fix the mess. Go home, enjoy your weekend/get a job."
Councillor Chris Vinson, leader of the Conservative group and the member for Walmer on Dover District Council said the protest appears to be "illegal migration-related", but said that the demonstrators do not appear to have placards or banners.
He told the Press Association: "Overnight, clearly what appears to be several hundred masked men wearing black clothing and balaclavas seem to have co-ordinated and loaded into rental vans and then descended on roads leading in and out of Dover with the intent to block them, and that's what they've successfully done, and it's brought the town to a standstill.
"There are stacks of people who are stuck in their cars or in their workplaces as a result."
He added: "I'm not aware that there's been any violence, although clearly, having the path of your vehicle blocked by men wearing black paramilitary gear and balaclavas is extremely intimidating, especially when it's not clear why they're there, and the police weren't aware and so haven't been able to warn people in advance and make sure that they are made to feel safe.
"I know people who have worked night shifts in the port who are being prevented from leaving for fears for their safety.
"Women who are in cars feeling intimidated, people stuck in traffic – who obviously are without toilet facilities – queues way back on the A20 and the other approach roads into Dover."
A local Reform UK councillor, Paul King, told residents to avoid the port area. He said: "I've been told it's a stop the boats protest. Speaking to local women, they felt frightened by such a large group of men walking up the street. We don't condone causing public disturbance by protest. But it's a very good example of what is happening daily with the boats coming in."
The shipping company DFDS said: "We are aware of the current traffic disruption affecting access to the port. Rest assured we will accommodate you on to the next departure once you arrive."
P&O Ferries said its sailings were still going ahead as scheduled, adding that those who missed their departure would be offered the next available crossing.