Is it possible to buy another 16-25 Railcard before my current one has expired?

Yes you can do that, but you need to create a new account. From the FAQ:

Providing the expiry date of your Railcard is within 30 days you will be able to renew it. Just log back into your account. You can renew your Railcard if it is due to expire before your 26th birthday.
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You will not be able to renew a Railcard if its date of issue is after your 26th birthday unless you are a Mature Student.

However, you can still buy another 16-25 Railcard by setting up another account using an alternative address and making your purchase just before your 26th birthday.

We will not be able to refund you for any over-lapping period of validity between the two Railcards.

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

http://www.16-25railcard.co.uk/eligibility-benefits/where-to-buy/

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.

(My bolding)