Is it worth/okay to write an email to a grad school I'm waiting on in an attempt to strengthen my position as an applicant?
I think David's comment sums it up well:


  Emailing the admissions office will almost certainly do nothing. Emailing a professor you'd like to work with could be useful[.]


Regardless, emailing is a complete shot in the dark -- the more likely possibility is your email is ignored. But it depends on how you write the email, and how impressive the additional internship is.

For best results, I would not phrase it as a way to boost your application, but rather just an email to say "hi" and introduce yourself, and ask about the status of your application. But then you can find a way to throw in the additional qualification somewhere.