By default, project managers have permission to create a case within the project(s) they are a project manager of.
You can give specific project reporters the ability to create cases too.
Steps
Go to the Project Preferences page of the project that you want project reporters to be able to create cases within
Confirm that the Enable cases checkbox has been ticked (see article on how to turn on cases if the checkbox is not ticked)
In the who else can create cases text box, enter the names of the project reporters whom you want to give permission to create cases
Scroll to the bottom of the page and press save
Now that you have done this, those users will see the New Case button when they are on the profile page of a contact/client who is part of a project that has cases turned on.