Poets who wrote during the reign of James are called "Jacobean." Sorry for the missing clause.
Elizabethan poets refer to those who lived or wrote during the reign of Queen Elizabeth I (1558–1603). Those who lived or wrote during the reign of her successor, James I of England who was also James VI of Scotland, (1603 – 1625) Since a poet's life might span the reign of more than one monarch, it is sometimes more accurate to use the term "early modern English poets" (or British, depending in whether any were from the non-English parts of the British Isles.)
To find a comprehensive list of English poets of the period, you should do a search of union catalogues listing primary publications. For example, you might search COPAC, the union catalogue including most British university libraries as well as the British Library, using the main search interface to limit dates and select poetry. Alternatively, you could find texts in the Early English Books Online (EEBO) database which can be found via the electronic resources page of most university library catalogues.
A short cut would be to go into the main catalogue of your university library and enter in the words "Elizabethan poetry" as first a keyword search and then a subject search. The search results will include many individual Elizabethan poets and also anthologies of Elizabethan poetry containing the "greatest hits" as it were of the period.