I submitted my AMCAS app in September (secondaries done sometime before mid-November) and had decent luck during the application process. However, I can't stress enough that I was very very lucky that things worked out so well. I was not an amazing applicant (my numbers were low to average for MD/PhD), and I honestly didn't expect to do as well as I did applying so late.
If you have a good, solid application, you will probably be fine - but if you don't (i.e. if you are average) you may have more trouble than you expect. Two of my schools rejected me immediately because they had given out all of their interview invites before they even looked at my application, and several others told me that I made things very difficult for myself by applying late.
If you are taking the MCAT in September, your situation will probably be even more difficult than mine was. Not impossible, but far from ideal.