Yes. You can buy it at the DB sales office at FRA. But you need a photo with your application. (There are automated photo booth machines at the airport for this purpose). You will then receive a temporary card (that looks like a train ticket) with your BahnCard information on it. It will not be possible for your friend to use it. As the permanent BahnCard has your photo on it.

Also, when you are at the DB counter purchasing the card, make sure you tell them to cancel it after the year is up. It is an automatic renewal system. So otherwise, you will get sent a new one with a bill for payment. Just have them handwrite your desire to not renew for another year and then sign it. They will then attach it with your application for processing.

BTW, how much train travel do you plan on doing? It might not be worth it.