I don't think this is possible without Admin right.
AR System Administrator Console is accessible only for Admins group and no other user can access it.
License can be released after certain period of unactivity or an Admun can release a license if required.
Its given un the ARS Config Guide 7604. Page 54
You will require good amount of customizations to achieve this, no OOTB method is available to achieve this. Only Admin uses can access licensing related forms.
May be you can try to add an Entry point for License Review Form so that it is visible under Quick links section of Application list on Home Page. You can then play with Roles/Groups to give specific permissions to this form so that your target users can access this form from Home Page directly. Try it, I am not sure if it will be allowed by AR System because there may be lot of other validations present on performing the actions on License Review Form.
you can provide only "Licensing " permission,,,
If you provide only "Licensing" permission, you can only modify the license "AR userr" as describe : "Licensing permission grants access to assign, modify and delete Action Request System User Licensing."
It will not allow you to add application permission (you can only with Contact Support Admin).
Specifities: to give Security you need to be security and Administrator for Administrator