If my father just had a mild stroke, can he still have the operation to clear out his carotid artery?

Yes. Yes, he can have a surgery if his carotid arteries are shown to be blocked. The operation is a carotid endarterectomy but is not offered if the carotid is 100% blocked.
Yes. He needs an MRI to see if the stroke has healed, usually 4 weeks. Once it has healed, he can have endarterectomy or carotid stenting with protection device. If symptoms keep coming back, he may need the procedure urgently. At my hospital this is decided by a team made up of neurology, vascular surgery , and interventional radiology, with patient input of course.