The woman was shot amid a standoff between law enforcement and supporters of President Donald Trump at the building in Washington DC. Her death was confirmed to the US media.
Earlier reports said she was in a critical condition.
The circumstance of the shooting, and if law enforcement were involved, is not yet clear.
Two improvised explosive devices have been safely seized by the FBI.
Several other people have been injured during the protests.
Thousands of Trump supporters swarmed the building following a rally close to the White House this morning.
It led to Congress members being escorted out to safety by police.
Robert Contee III, Washington DC’s Chief of Police, has said “chemical irritants” were used by demonstrators to access the building, according to Associated Press.
READ MORE: Trump on verge of COLLAPSE as Twitter locks his account
Mr Biden wrote on Twitter: “Let me be very clear: the scenes of chaos at the Capitol do not represent who we are.
“What we are seeing is a small number of extremists dedicated to lawlessness. This is not dissent, it’s disorder. It borders on sedition, and it must end. Now.
“I call on President Trump to go on national television now to fulfill his oath and defend the Constitution by demanding an end to this siege.”
Mr Trump told protesters to “go home” in a recorded statement on Twitter.
He said: “I know your pain, I know you’re hurt.
“We had an election stolen from us…but you have to go home now, we have to have peace.
“This was a fraudulent election but we can’t play into the hands of these people but we have to have peace.”
President Trump’s accusations of election fraud have been rejected by independent observers and US courts.