Campaigners representing European nationals said they had received hundreds of complaints of people being denied a vote even though they are on the electoral roll.
To be able to vote, EU citizens had to complete the EC6 or UC1 forms to declare they would not cast a second ballot in their home country, and submit them to their local council by 7 May.
The Electoral Commission said the “very short notice” that the UK would participate in EU elections meant it had not been able to properly publicise the extra requirement.
However, many reports circulated on social media of EU nationals who had emailed or posted their forms before the deadline but were still denied a vote, or who were promised a form would be sent that never appeared.
University of Edinburgh student Anika Edrei, 24, received an email from Edinburgh Council promising a form would be sent, but says it never arrived.
“This is absolutely unconscionable,” the Finnish national told the Scotsman. “This election is a kind of second referendum on EU membership, so I was very invested in being able to vote, and I’m really angry to be disenfranchised in this way.
“There is nothing I can do to vote today. When I called [the Lothian Valuation Joint Board, which handles electoral registration for Edinburgh], the man I spoke to was very upset and angry, and said it was completely undemocratic. He was upset he couldn’t do anything more to help me.”
Campaign group The 3Million called for an "urgent investigation into this democratic disaster".
In a post on twitter, Ms Sturgeon wrote: “Just spoken to a constituent at a polling station who is from Poland, been here for years but wasn’t allowed to vote… even though he’s on register. It is outrageous.”
She urged EU nationals who could show they had completed a form to contact their local authority before polls close.
The SNP’s Westminster leader Ian Blackford tweeted: “It’s heartbreaking to see reports of so many EU citizens denied their right to vote today, all because of the incompetence of this Tory government.
“EU citizens are a fundamental part of our social fabric - their right to vote should be undeniable.”
The issue was raised at Prime Minister’s Questions on Wednesday by SNP MP Joanna Cherry, who said the government should make forms available in polling stations to stop people being disenfranchised.
Theresa May replied the lack of notice for votes was “because of a decision by this House on 29 March not to agree a [Brexit] deal that would have made it unnecessary to hold European elections”.
After PMQs, the Prime Minister’s spokesman said the government was “aware of the issue” but could not offer any action that had been taken to address the problem.
“I have been shocked, but sadly not surprised, by the reports coming in from across the country of electors being told they cannot take part in today’s vote,” Ms Cherry said.
“Yesterday at PMQs I asked Prime Minister to user her office to sort this out and make sure the UC1 form was available at polling stations.
“Her high handed and dismissive response was typical of the Tory Government’s approach to the rights of EU Citizens.
“It is particularly egregious to hear of some Irish citizens who do not need to complete this form at all, are being are also turned away. This process has gone badly wrong.”
In its statement, the Electoral Commission said it had called for changes to the registration process for EU nationals after the last European election in 2014.