If you want to restrict an employee as to who they can submit their expense entries to (or even time entries), you should un-check the security permission Show submit options when submitting expense entry under Expense Entry in their Security Profile. Check CORE Help Center for details on how to customize the security permissions.
After you do this, CORE does not prompt the employee with the Submit dialog on submitting the expense entry, but the expense entry is submitted to whoever is specified in the Employees > General > Rates & Options screen under Submit To by default. You can also choose to submit to the Employee Manager, Client Manager, Project Manager, or Specific. In particular, if the option Specific is selected, the entry is submitted to the Project Manager. as it is a required field and so always has a value.