Where can I get dragon bones and scales without killing dragons?

Solution 1:

No, killing dragons is the only source of dragon bones and dragon scales. If you roam around the world enough you'll find dragons, there are unlimited dragons available to spawn according to Todd Howard, the director.

Solution 2:

To find dragons, you need to do some main quest missions.

Killing the dragon in the tower is enough. You can also try to go to dragon stones, or do a lot of teleports to villages and towns, as dragons usually appear when you teleport (higher chance than normal to appear. but not much (not confirmed)).

You don't need to kill many dragons for the armor because one dragon gives enough materials for one equipment piece.

Solution 3:

I don't think you can get bones without killing a dragon, but there is one place where you can get scales. On the hill north/northeast of Rorikstead, there is a shrine to Akatosh that has a few "offerings" lying around it. One of these is a dragon scale. If you leave the area and wait for 10 days, the items will all respawn, meaning it is possible to craft and upgrade a full dragonscale set before doing the Dragon Rising part of the main quest. If you have the Dawn Guard DLC you will find one dragon bone in the laboratory of Serana's mother.