Can my coworkers have their own account and have access to a form I've created?

Yes, you can make them a sub-user of your account. Please check this guide: https://www.jotform.com/help/232-How-to-Share-Forms-with-a-Sub-Account-User.

Please take note that under the Free plan, you can only have 1 sub-user. You need to upgrade to a higher plan to have more sub-users.

Hope that helps. Thank you.

Thanks for the quick response, Jan!

Could I get a free trial of the upgrade version?

Free accounts can add one sub-user.  It is possible to test the functionality with the free sub-user and then upgrade later if you need more sub-users.