Hey Hannah,
This should be possible by adjusting one of your WordPress core settings. To enable anyone to register on your site, and therefore submit events, you should first head to “Settings > General” in your admin area.
Once there, check the “Membership” option so that anyone can register – here’s screenshot of this options → https://cloudup.com/c5n96djltqH
Once that is checked, people should be able to register for your site by navigating to /wp-login.php?action=register on your site (so, eventsboxuk.com/wp-login.php?action=register), and see a signup form like this → https://cloudup.com/cBzMaBG9URi
You can then place a “signup” link anywhere on your site, even in a navigation menu or something, to eventsboxuk.com/wp-login.php?action=register so that people can sign up.
I hope this helps! Let me know if it does, or if you have any other questions/concerns.
Cheers,
George