An administrator asked if it is possible to limit the number of outgoing Internet emails each user can send each day.

Yes, Exchange 2010 and 2013 can limit the number of messages a user sends, using the RecipientRateLimit and MessageRateLimit parameters in a Throttling Policy. These policies apply to both internal and Internet email.

MessageRateLimit controls the number of messages per minute that can be submitted. When messages are submitted using Outlook or OWA, the messages will stay in the Outbox longer when the user submits messages at a rate greater than the MessageRateLimit parameter. The messages are deferred, not denied, and will eventually be sent.

RecipientRateLimit limits the number of recipients that a user can send to in a 24-hour period. The user will receive an NDR for messages in excess of the limit.

Distribution groups (including dynamic distribution groups) in the Global Address List count as one recipient.

To create a new policy where the users can send to 30 recipients a day and no more than 1 message per minute, you would use this command:
New-ThrottlingPolicy -Name LimitMessagesSent -RecipientRateLimit 30 -MessageRateLimit 1

To assign it to a user, use this command:
Set-Mailbox -Identity user_alias -ThrottlingPolicy LimitMessagesSent

More Information

Understanding Message Throttling
https://technet.microsoft.com/en-us/library/bb232205.aspx

New-ThrottlingPolicy
https://technet.microsoft.com/en-us/library/dd351045.aspx