It looks like if you have already submitted your application and it has been verified already, schools should start receiving the verified apps from AMCAS starting on about July 8. Depending on the schools, you could start getting secondaries immediately after that, up to a month or more after that.
In general, you can start receiving secondaries as soon as a week or two after submitting your app, depending on how efficient AMCAS is at verifying your app and transmitting it to the schools, and then how efficient the schools are at getting stuff out to you.
For me, I got my first secondaries within about two weeks of submitting the app, and my last secondary came at the beginning of January, which was more than 2 months after I submitted. Yes, I admit it, I applied REALLY late last year (end of October).
Basically, you never know. Just be prepared.
Good luck! 😀