OK,first, we are knowledgeable about this. You can camp at The Tetons in Sept inside the park, we did 10 days a few weeks ago. Check Rec.gov and see if you can book spots. We had to move three times in that ten days.
Otherwise your best bet is Shadow Mountain NF dispersed camping. A very good camp just outside the park past the Mormon Village area. But there are only 15 or so spots there.
Otherwise there is a cg at Slide Lake past shadow mountain to the east. Its rather far out and the last mile or so of road is rough.
There is another dispersed CG at s cg at something lakes near the Triple X Ranch this is a dangerous spot, if its been raining you will get stuck.
For YNP, about 17 miles west of Gardiner is the BLM dispersed CG called Carabella. It is marked a fishing access area but there are 15 or so good spots on the on the Madison River. Good cell phone signal there and at Shadow Mountain. Just to the south, 15 mi from Gardiner is USNF Canyon CG which generally has open spots in the morning. Not a great spot, no cell signal and the road is rock strewn and rough and narrow. We used it will waiting for an open spot at Carablla.
O Outside+- the Cody gate there are several nice NF CG. No cell signal. Around Island Park, west pf West Yellowstone there are many NF dispersed cg.
We have done a lot of camping at these two parks the last few uears. And we got got in Bridge Lake cg in YNNP for five days this year rethought after the Tetons. They are closed now.
The weather can get very bad this time of year.
Arctic Fox 25Y Travel Trailer
2018 RAM 2500 6.7L 4WD shortbed
Straightline dual cam hitch
400W Solar with Victron controller
Superbumper