Nearly 200 countries are in Busan to negotiate the deal, but there is little sign of agreement with just a day left before talks are due to end.
Greenpeace said the tanker Buena Alba, anchored off the Hanwha TotalEnergies complex, was scheduled to pick up propylene, which is used to manufacture plastic.
"The activists boarded the vessel peacefully and met no reaction from the vessel crew," said Greenpeace spokeswoman Angelica Pago.
"We painted 'PLASTIC KILLS' on the side of the vessel and the climbers successfully set up a camp," she told AFP.
"They intend to stay in order to continue putting pressure on the negotiators to resist fossil fuel and petrochemical industry interference in the talks and to deliver a treaty that firmly cuts plastic production."
A spokesman for South Korea's coast guard told AFP police had been "deployed on the ship, and we are making warning announcements to facilitate a safe disembarkation".
He said a "thorough investigation" would be carried out to determine if there were any "illegal elements" to the protest.
Attempts to reach an agreement on curbing plastic pollution have stalled over several key sticking points, including whether to cut new plastic production.
Dozens of countries, backed by environmental groups, insist a treaty without production cuts will fail to solve the problem, but a group of largely oil-producing states is fiercely opposed.
"The brave activists that boarded that vessel today show the courage and should inspire governments here to hold the line and do what everyone knows is obvious," Greenpeace delegation head Graham Forbes told reporters in Busan.
The negotiations have reached a "pivotal moment", he said, but "a handful of governments... are looking backwards and refusing to take the steps necessary for us all to advance".
"I think we are at a very risky moment right now of being sold out, and that would be an absolute catastrophe," Forbes said.
sah/roc-kaf/rsc
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