Can I obtain all DIY recipes without venturing out to other islands?

Solution 1:

This answer will probably get outdated quickly as more information is discovered, but here goes.

You get one DIY recipe daily from a message in a bottle that washes up on your beach. You also have a chance of getting them from flying presents.

Nook and his offspring will give/sell you most of the "essential" DIY recipes throughout your stay on the island.

You can also get a DIY Recipe from an islander if they are in their house working on something when you speak to them.

It's likely that nobody knows how many recipes there even are or their sources. That said, you will always have a steady stream coming into your island without "outside help".