"So...am I able to get a permit for the JMT at Whitney Portal?"
As others have stated, you can't get anything at Whitney Portal except for maybe a parking citation. The wilderness permits all come from the Forest Service at the Interagency Visitor Center that is a mile south of Lone Pine. When I was there last Monday, there was a waiting line of people for just such a permit. I have no idea how successful they were. It is a crapshoot.
As for the remarks about needing oxygen on the way up the Switchbacks, I hear you! When I went up there last week, I was carrying only about 12 pounds. Nevertheless, the thin air will catch up with almost everybody. I graphed out my results since I was using a pulse-oximeter to gather data. This graph is complicated, so you have to study it a bit. It told me that I am not getting any faster.
Going up the Switchbacks is part of the problem. Then when you reach Trail Crest, the cool breeze from the west hits you. That tends to chill you somewhat on the way down the two-tenths of a mile to the JMT junction. Of course, most hikers are going up to the summit from there, so they drop their backpacks temporarily at the junction. Three marmots live there in the talus, so they come out and have their way with your backpack. I recommend temporarily hanging your pack at eye level off the rock cliff right there.