Is it appropriate for me to contact the company as an intern about this, and if so, how?
Perhaps a more professional approach would be for this person to apply on it's own, and you agreeing on being included as a reference or endorse him in other ways after he applies.
In a way it could be ok to directly recommend him, but it is also a gamble, as you could bias or affect his application if your boss or the person you contact takes this as you trying to influence their decision.
If you care for this person, you can help him by probing around and find if they are currently seeking for more interns. If they do, you can inform him so he can apply by his own means, after which you can give faith on this person's work.
As a related reading, you might consider checking my answer and the other's to How early can one refer a friend/family to a job after starting a job at that company? Although in that case the person only had 2 weeks working there (and the person was his brother), the same suggestion I gave works here in a different degree:
Now, the situation you have here I consider is a bit different than the way you are focusing it. I believe that given you only have 2 weeks working there probably any referral or recommendation you give now will not be regarded as highly as the ones of senior or more experienced coworkers.
In this case I say you recommendation will be taken more seriously that the one of a two-week worker, but still it could not be much effective and even affect your friend's application if you company feels you may be trying to influence their decision.
Let him apply on his own and recommend him when he does and advances on the recruiting process.