As an Admin, you can go into their mailboxes, as you put it, by
resetting the user password. What you can't do is go on fishing
expeditions without letting the user know, and getting their
agreement.
You need to be careful about privacy. Google clearly take that
seriously and there are specific obligations on you, as Customer, in
the Education Terms
http://www.google.com/apps/intl/en/terms/education_terms.html,
copied below:
2.4 Privacy. Customer agrees to protect the privacy rights of its End
Users under all applicable laws and regulations. Customer’s
Administrators may have the ability to access, monitor, use, or
disclose data available to End Users within the End User Accounts.
Customer will obtain and maintain consent from all End Users to
Customer’s access, monitoring, use or disclosure of this data, and to
Google providing Customer with the ability to do so. Customer is
responsible for obtaining any necessary authorizations from End Users
to enable Google to provide the Services.