You will have your MCAT score by end of July if you take it in June. While it's by no means late, I would suggest having all your transcript/LORs/PS ready by then and ready to submit as soon as the score is released. July is when the AACOMAS starts picking up on volume, and it may very well take at least one and half month for your primary application to be verified. You are realistically looking at early September when your primary application is released to each school.
If you are consistently scoring 30+ on mock test (or the equivalent percentile), you could opt to take the May MCAT should a spot be open closer to the test date.