I believe they close at 5 PM, but you need to be there by 4:30 to pick up a permit.
You can call the Wilderness Permit number (760-873-2483) and arrange a night box pickup at the kiosk in front of the Visitors Center.
Are you day hiking Whitney? If you do a night box pickup and then drive up to the Portal, it sounds like you aren't giving yourself any time to acclimate to the altitude. Would like to hear your plans.