When is the black fly season in Newfoundland?

Whenever it's warm enough to go outside without a parka lol!
Seriously, I assume you're planning hiking/camping. It's usually worse in June on the island. But we're having one of the coldest springs, 5C today--the HIGH, not the low-- so everything is probably going to be late.
DEET, my friend.

We were actually thinking of September.

September is actually a great time as long as you don't want to lie on the beach. Normally (assuming this cold rain ever stops & we go back to normal weather!!) September is 10-18C, dry & sunny, with light winds. And the black flies have normally gone--YEAH. A good many tourist locations have started staying open through September, but it will be worth your while to check if you have a specific park/B&B/museum in mind.

5C today--the HIGH, not the low
Wow. In Victoria the low is predicted to be 10C. I registered 25C today and the prediction for tmmorow is 28.

We have the plagues of black flies and tourists to acompany the weather though.