Yes, you should do it in SQL. I'd use something like (untested, I don't have a database available at the moment):

CREATE TABLE employee (    -- Lists availability times
    ID       NUMERIC,
    start    DATETIME,
    end      DATETIME
);
CREATE TABLE scedule (     -- Lists scedules appointments
    ID       NUMERIC,      -- Foreign key into employee
    start    DATETIME,
    end      DATETIME
);
CREATE TABLE appointment ( -- Unsceduled appointments
    ID       NUMERIC
    start    DATETIME,
    end      DATETIME
);

SELECT employee.ID, appointment.ID FROM employee, appointment
WHERE  employee.start <= appointment.start AND  -- Check availability
       employee.end   >= appointment.end   AND
       employee.ID NOT IN (  -- Exclude employees who 
                             -- already have an appointment
             SELECT scedule.ID FROM scedule, appointment
             WHERE  scedule.start <= appointment.end AND
                    scedule.end   >= appointment.start
       )
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