There may be a barrier to doing the purchases online using your existing account
But as it says on this page http://www.16-25railcard.co.uk/eligibility-benefits/eligibility/
You can purchase whilst 24.
My recommendation would be (and what I did when I was a bit younger) to make the purchase in person at a railway station. And to purchase it as if it were your 1st time (ie take all the documents requires)
When you purchace at a railway station the onus is on you to the register the card online, but there is no obligation to
This is detailed on this page of the website
At a staffed station ticket office
You can buy your Railcard at any staffed station ticket office and some National Rail-licensed Travel Agent. You can do this on the day you are travelling.
If you're applying at a station for the first time you'll need to bring one of the following:
Your birth certificate
Passport (all nationalities accepted)
UK driving licence
National identity card
You'll also need:
A passport-size photo
If you're renewing your Railcard, you'll need to fill in the application form, and take your existing Railcard and Photocard. These are automatically accepted as proof of your age. If you are renewing an online Railcard at a station you will need proof of age as above.
Once you've bought your Railcard at a station, you can then register it online. This will make it easier for you to renew or buy other cards online.