The application is the primary AMCAS app that has all your personal information, coursework, grades, and personal statement. That can be submitted as early as June 2 and schools receive them as early as June 24. Then schools send secondary applications which may require additional essays (and $$$). Sending letters is part of the secondary app.
If you take a June MCAT you won't get your scores back until July which isn't too bad. If your premed committee doesn't have your letter until August that's starting to get late but isn't absolutely terrible. You would have to have everything else done, written, paid for, and submitted so the letters would finalize all your applications so you could be considered for interviews.
I find it rather strange that your committee wouldn't send a letter until August. Most require everything to be done by May so they can get letters out on time.