Use a database table Authorization as shown below with appropriate choice of data types.  This table must be populated by you manually using the PhpMyAdmin interface to MySQL.  The address should be a primary (hence unique) key of appropriate type.   The limit on the number of emails sent to the address should be initialized to some small limit (which can be different per address); the current number sent should be initialized to zero and incremented at each approved request until the limit for the address is reached (based on an SQL update command).  It can be reset manually.

Authorization (for email requests)

The value of the input address is obtained from the form and the restriction on its number of emails requires using PHP and SQL to compare a pair of columns in the table.  My NJIT address has to be included as above so I can test your work online by sending an email to myself.