Britons aged 26-30 are eligible for the discount card on a first-come, first-served basis, the Rail Delivery Group (RDG) said.
Many passengers took to social media to express their frustration at being unable to use the website selling the cards.
Emily Thomas, from London, wrote: “Getting a 26-30 railcard is worse than getting a Glastonbury ticket. Sort out your website to make it fair!”
In a message to the National Railcards Twitter account at 7.48am, youth worker Josh Booth, from Leeds, posted: “Been trying to get 26-30 railcard since 7am, your website crashing and been on hold on the phone since then.”
Tom Drury, from London, wrote: “The launch of the new railcard is a shambles.”
A limit was placed on the number of cards available while the scheme is trialled to assess the impact on revenue and passenger numbers.
It costs £30 each year and saves passengers a third off most fares.
The card is aimed at leisure travellers, with no discounts on season tickets and a £12 minimum fare on all journeys between 4.30am and 10am, excluding weekends and public holidays.
It must be bought online and downloaded on to a smartphone.
The card was previously only available to a limited number of people in East Anglia.
Chancellor Philip Hammond said in his Budget speech in November that the railcards would give “4.5 million more young people a third off their rail fares”.
There was previously no nationwide railcard available for people between the ages of 26 and 59 travelling alone unless they were disabled or in the armed forces.
An RDG spokesman said: “The trial of the 26-30 Railcard is part of the rail industry’s long-term plan to change, improve and boost communities by enabling more people to travel by train.
“Research being gathered as part of the 26-30 Railcard trial is being used to inform discussions with the Government about a national roll-out, and to develop new products that make leisure travel easier and better value for customers.