The process of getting a walkin permit is somewhat convoluted.
Im sure Kevin can answer your question better than I but I'll give is a shot because I just went through it.
Between May 1st and Nov 1st there is a maximum amount of people (quota) that are allowed in the Whitney Zone (which includes Mt Muir, Whitney and Russell)
During this quota period 100 day hikers and 60 overnight hikers are allowed in the Whitney Zone daily.
There are 2 ways to obtain a permit during the quota period. (Non quota period you can get a self issued permit outside the visitors center.)
1. Try the lottery
2. Walkin at the Lone Pine Visitors Center
All lottery winners must pick up their permits at the Lone Pine Visitors Center by noon the day before their hiking date. By 2:00pm the FS knows how many permits have been issued for day and overnight use for the next day. If you go to the visitor center after 2:00pm they will issue you a walkin permit for any lottery winners that did not pick up their permits by noon.
Now lets say you check the
calendar of dates for day use of Whitney Zone and find that Sat. Sept. 3rd is not full because either the lottery did not get 100 applications or there were cancellations. If you go to the Visitors Center after 11:00am the day before they will issue you a permit for the 3rd.
Lets say when you show up at 11:00am there are 20 other people that have shown up trying for those 3 permit. If this happens they will hold a lottery by picking numbers out of a hat. Lowest number gets first choose. If you happen to pick #1 and there are 3 people in your team than everyone else is SOL.
All walkin permits issued are free.
Bottom line: show up at the visitors center at 11:00am and ask if any permits are available. If not, show up at 2:00pm and ask again. If you strike out show up the next day and try again. You shouldn't have any problem in Sept.
I got lucky this year in July due to a high snow year. There were a lot of cancellations.
My first walkin was on Sunday at 2:00pm and there were 9 permit available for Monday. The second time I tried at 11:00am and there were 6 permits available for 15 people that were waiting there. They held a lottery and I picked #2.