The US-based organisation said the Trump administration was a "firm reminder" that it needed to adapt.
The organisation archives 300 million websites every week, but said it was preparing for "a web that may face greater restrictions".
But it has admitted that storing a copy in Canada would cost "millions".
The Internet Archive is a huge record of websites, books, audio and software that can be browsed free of charge online.
It lets anybody look up material that was posted online, and see how a website has changed over the past 20 years.
The non-profit organisation said it wanted to keep its "cultural materials safe, private and perpetually accessible".
It said storing a copy of the data outside the US would mean "no-one will ever be able to change the past", although the Internet Archive does let website owners have their pages removed from the database.
A copy of the data stored in Canada would not be subject to new data or censorship legislation introduced in the US, if the Trump administration decided to introduce any.
Mr Trump has previously suggested areas of the internet should be "closed off" to tackle extremism.
But it has been suggested such a move would violate the First Amendment of the US Constitution.
"Any attempt to filter out the online activities of extremist groups would inevitably infringe on the First Amendment rights of Americans," David Greene, of the civil liberties group Electronic Frontier Foundation, said in 2015.
"Even if you would accept the proposition that some of this speech is illegal, it's impossible to block just that out."
Jim Killock, from the UK's Open Rights Group, welcomed the Internet Archive's announcement.
"Libraries in the UK should remember that they too are subject to excessive surveillance powers that may be abused, and UK internet surveillance material is accessible to the NSA [National Security Agency]," he said.
"Given the surveillance capabilities of the US government, this is a sensible and practical move to preserve the Internet Archive."
The Internet Archive, which stores copies of billions of websites, says it will keep a back-up in Canada following Donald Trump's US election victory.