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Instagram's new API is open to regular users as well (that's what Tweet Photo uses), but I believe it's read-only.

Zapier etc probably have special deals with Facebook.

Thank you! Last time a looked at it a few years ago I couldn't make heads or tails of it. I'll take a look again.

There’s an approval process you need to go through. I wouldn’t recommend it if you can avoid it as you’ll be at the mercy of Facebook.