The standard line (which this year was not adhered to) is that AMCAS takes 6-8 weeks to verify your application. Then they submit the app to the schools who then send secondaries.
If that holds true, the schools might start sending you secondaries at the beginning to middle of August. As was stated earlier, the schools each have their own unique way of doing things. Some are all online, even paying the fee, some are online, but you mail the check, some are all pony express and some e-mail you the secondary, but then you pony express it back to them finished, etc.
What is more, what the schools did this year might change for next year! I have heard rumors that some schools may ditch AMCAS because of this year's ongoing debacle. That would put everything I just mentioned in turmoil as well.
Bottom line, June-July travel would be best from what we know now, but if you are a strong applicant, a couple of extra weeks might not hurt that much, but some secondaries have deadlines.