This is Joomla ACL. The best thing I can do is tell you to read our documentation and the linked tutorials in there. I can only tell you where to look for access controls but I can't offer a solution since it depends on your users, user groups, viewing access levels, menu items, categories and permissions settings. This is all choices you've made yourself about.
It's possible that one or more of any of your menu items, including those in menus not published anywhere in the front-end of your site, pointing to the Akeeba Ticket System all categories or a specific category has an Access that's not fulfilled by logging in, e.g. Guest, Special and so on. You have to check all of your Akeeba Ticket System menu items and make sure that their Access is correct.
Further to that, you should check the Access of all of your Akeeba Ticket System categories. They must be accessible by your logged in user.
Finally, check the Permissions for each category (Akeeba Ticket System, Categories, click a category, Permissions) AND for the overall component (Akeeba Ticket System, Options button in the toolbar, Permissions) to make sure that the user groups of your user allow the creation of new tickets. If you can see the tickets of a category and the new ticket page but NOT file a ticket (I think that's what you are saying) then this is exactly what you need to check first.
Nicholas K. Dionysopoulos
Lead Developer and Director
🇬🇷Greek: native 🇬🇧English: excellent 🇫🇷French: basic • 🕐 My time zone is Europe / Athens
Please keep in mind my timezone and cultural differences when reading my replies. Thank you!