Can an employee's expense requests only go to their manager?

Yes, you can customize who receives notifications about expense requests from employees. To do this: 

1. On Remote, go to the Company settings tab.
2. From the Company information tab, scroll down to the Notifications section.
3.  Click on Edit.
   
   
   
   
4. From the modal, choose which users should receive expense notifications, then click on Save. "Managers" in this case, represents the employee's People Manager. If an employee hasn't been assigned to a People Manager, their expense notifications will go to the Company Admins. 
   See also: How to assign a People Manager to specific employees