For tournament organizers, an invitation can be resent by simply clicking on the exclamation point icon next to the invited participant:
Once clicked, the participant should receive a new invitation email within 5 minutes.
For tournament participants, It is possible to recover an invitation from the account's notifications. Once logged in to Challonge.com, navigate to the 'Notifications' page from the main menu. This page will contain the same invitation message sent via email: