Another suggestion if you are prepared to be a bit bold. Find a paper (preferably published as recent as possible) that especially interests you and is written (primarily) by a researchers about half-way in the hierarchy in the institution. Then try calling that researcher. You should say something along the lines of:  
"Hi I am so-and-so, a student at stage X of my degree Y. I am very interested in your research on subject Z, especially your paper A. I was wondering if you have any positions for students like me for over the summer. Can I e-mail you about this or should I e-mail someone else?".  
In my experience, almost always they will say: sent me/that person an e-mail on this e-mail address. Of course you have your e-mail ready to go, so you hit send as soon as you hang up.  
This strategy has several advantages:  

 1. It humanizes you. An e-mail can be sent to dozens of people with
    fairly little effort, a call takes more commitment. Or, it at least
    feels like more focused. It also shows you are interested in the
    actual research, instead of just the name of the top professor and
    the institution.
 2. Usually e-mails get read sporadically and e-mails from outside students get very low priority or at times forgotten. After this call the researchers knows to expect an e-mail at a specific e-mail address about this specific topic. If you are lucky the researchers gets curious and looks at it right away.  
 3. If there is no chance of getting a position over the summer or you need to approach a specific person to get a chance, you get to hear it right away.  

Check their website first if they have any policy on it for students wanting a position like you are looking for. If they do, follow that protocol. If not, the above protocol is fair game.

If there is no published e-mail of the person you want to contact, contact the general e-mail of the institution and ask for the researcher you want to contact. You might get a secretary that ask you to mail, in that case just mail. 

I expect academics to disagree with this advice. They don't want to be called by students and want to let requests like this linger in their inbox for days or weeks until they have a moment. However in my experience calling works anyway, and academics prioritize people who called over people who only mail.