Just 10,000 of the railcards - enough for one in 500 of the eligible population - were released by the Rail Delivery Group (RDG) at about 7am, on a first-come, first-served basis.
There are 10,000 of the 26 to 30 railcards now available on a first-come, first-served bases.
It is thought that demand for the tickets is particularly high because only 10,000 have initially been made available as part of a limited trial of the scheme.
Many people turned to Twitter to share their frustration at being unable to get on the website, with #railcard trending on the social networking site.
A nationwide trial of the so-called "millennial railcard" descended into chaos this morning, with the mad scramble to secure the discount cards causing the website to crash. "Please bear with us, we've got the whole team working on it and hope we can have it back up shortly". "Sort out your website to make it fair!"
It costs £30 each year and saves passengers a third off most fares.
Meanwhile, Tom Drury, from London, wrote: "The launch of the new railcard is a shambles". There will be a £12 minimum on tickets between 4.30am and 10am, excluding weekends and public holidays.
A card offering discounted rail fares for 26-30-year-olds will launch nationally, but only 10,000 people will be able to get one.
Some people have questioned why there is a need for another trial of the card, after a trial across the Great Anglia network proved successful.
'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'.
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