The protesters evaded security guards just before dawn to begin the unauthorised bid to climb the 72-storey glass-fronted building, which towers 301 metres (1,017 feet) over the British capital.
Greenpeace said the six "artists and activists" had targeted the Shard to highlight the work of Shell and other oil companies and intended to hang an artwork from the top if they managed to reach it.
"They chose to climb the Shard because it towers over Shell's three London offices," it said in a statement.
Greenpeace said they were "free climbing" without assistance but would attach safety ropes as they progressed, while a live video feed of the climb was being broadcast online.
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The group named the climbers as "Sabine, Sandra, Victo, Ali, Wiola and Liesbeth" and posted a photo of them before the climb.
It quoted climber Victoria Henry, a Canadian woman living in London as saying before the climb: "It's going to be really hard work, it's going to be nerve-shredding for all of us and we may not succeed, but we're going to do everything we can to pull it off."
"We have officers down there, monitoring the situation. The event is still ongoing."
A Shard spokesman said they were working to ensure the safety of the climbers as well as that of workers and visitors to the centre.
The Shard opened to the public in February and contains office space, a five-star hotel, restaurants and luxury apartments.
It was designed by Italian architect Renzo Piano and financed by the Qatari government.
Oil giant Shell has been involved in oil extraction in Alaska since the 1950s, and said in 2012 it had completed top-hole drilling for two wells in the Arctic, the first drilling in the region in more than a decade.