One table with 2 foreign key mapping same primary key
Try this

SELECT
    status,
    schedule_switch.Switch_Employee_ID,
    schedule_switch.Requesting_Employee_ID,
    empl_info1.FirstName,
    empl_info2.FirstName, //added the modified join select column
    DATE_FORMAT(schedule_switch.date, '%m-%d-%Y') AS switchdate,
    DATE_FORMAT(time_slot.date, '%m-%d-%Y') AS date,
    start_time,
    end_time
FROM work_schedule
    JOIN time_slot ON work_schedule.Slot_ID = time_slot.Slot_ID
    JOIN manager_info ON manager_info.Manager_ID = time_slot.Manager_ID
    JOIN schedule_switch ON work_schedule.Schedule_ID = schedule_switch.Schedule_ID
    JOIN employee_info empl_info1 ON schedule_switch.switch_Employee_ID = empl_info1.Employee_ID
    // modified your join
    JOIN employee_info empl_info2 ON schedule_switch.requesting_Employee_ID = empl_info2.Employee_ID
WHERE status IS NULL
ORDER BY switchdate;