Yes, it will depend on when in Sept you plan to visit as lodging starts closing down after Labor Day in the Park. If you cant get in-park lodging thru Xanterra (only book thru them) then West Yellowstone is a good base. If you can add days and include the sights mentioned by thbergs, all to the good.
If you do wind up being able to do Beartooth Pass, then I would do 2 in the Gardiner area plus nights at West Yellowstone.
If you are up to a long drive your first day, you can make it from Wenatchee to either Butte (484 miles, 7:45) or Bozeman (567 miles ~9 hours) the first night. Then the next day you could continue on I90 to near Billings, drop down to Red Lodge and cross Beartooth to enter the NE entrance. I would stay in Gardiner that night at least and maybe the next night so you could finish up at Mammoth and see other things along that route. This route would be totally weather dependent. The earlier in Sept the better for Beartooth.
Then if you are flexible enough with time (and money, lol) spend another 3 nights in West Yellowstone or if you can get something in the park, Canyon, Lake or Old Faithful. I would not stay at Grant Village which is too far south.
Getting to GTNP you would exit thru the south entrance, but you could stay in West Yellowstone to do that.
If you don't want to do Beartooth, then it is very easy to get to the West entrance from I90 and staying in West Yellowstone is fine for 5 nights (4 full days).
There are bears everywhere and they are hungry in September!
Pam
