In theory you can go through the IAS any time of year, but you'll likely find in practice that one or more of the people needed to help with the testing, assessment, and forming the team will not be available in the summer.
Not that I'm advocating you wait. One can find problems with almost any time:
Beginning of school year: "Why don't we wait and see how this year starts off."
Middle of school year: "We'll know more at the end of the year."
End of the school year: "You really should have started this process earlier, there isn't enough time left to complete everything."
Summer: "No one is available at this time, please leave a message after the beep."
I'll be happy to share our experiences with very end of school year discussions (not very worthwhile, but that was the school more than the timing), beginning of school year (with skip post-poned until the end of the year), and 2/3 of the way into the year.
For gifted children, doing nothing is the wrong choice.