It really varies. A bunch of very uptight people apply in the first hour of the first day, thinking it will give them an advantage. It doesn't. Just look at any school specific thread, and observe how people are surprised when people who submitted after they did receive IIs first.
Earlier is always better than later, but you really have all summer to get everything in before you bump up against possibly being at a disadvantage after Labor Day. Keep in mind, though, that's a soft deadline for getting everything in, primary, secondary, LORs, everything.
The first step is to get your primary verified, and that could take up to 6 weeks after you submit, which is why people obsess on submitting early. AMCAS builds in a little speed bump for you, by holding everything until at least 6/24 (i.e., the application opens on 5/3, but they do not accept submissions until 5/31, and will not transmit to schools until 6/24 at the earliest, assuming you are verified by then).
Bottom line -- even though you can start working on the primary on 5/3, you cannot submit until 5/31. Will that work for you? If not, you can probably wait an extra week or two and still be verified by 6/24, but it's not an exact science, as the queue will build from around a day at 9:30 a.m., or whenever they open the portal on 5/31, and will geometrically increase from there. Within a week or two the queue will be long enough that you will no longer be able to be verified by 6/24. Not the end of the world, for the reason I stated above, but still, if you desperately need to be in the first batch, you want to be in by the end of the first week of June, and even that won't be a guarantee!
After AMCAS gets slammed the first few days, things steady, and people apply throughout the summer. As I said before, you want to be complete everywhere by Labor Day. Submitting by the end of June should allow you to do that. Earlier is better than later, but your earliest possible submission is 5/31, not 5/3. Hope this wasn't too long, and helped!