It depends on what you are looking for on your kayak vacation. June and early July are generally the best months for seeing Orca whales. The islands tend to be less crowded but offer a greater probability of getting rain (or "heavy mist" as we locals sometimes say). July and August offer the best weather but water traffic increases and campsites can be more crowded.
If you visit in September there are fewer crowds and the weather is usually good but the days are shorter. Migratory birds are starting to come in, making it nice for birding. Orcas and bald eagles may not be as plentiful in September, although that equation can be highly variable.
On one of our trips in late September, L pod (the largest of the three resident pods of orcas) swam within 30 yards the group. Then, during the middle of all this excitement a mature bald eagle flew overhead. Exciting stuff! Needless to say our guests loved it.
Whichever month you choose to paddle in the San Juans you can be assured of having a great kayaking experience.