I would definitely use organic fertilizer on your lawn. I am a landscaper and I use Corn Gluten Meal on my lawn--it is great, helps control weeds, slow release. Even then, I would wash it in before your chickens get out on it. With your plants, you will want to mix the fertilizer in with the backfill around each plant. I have had a problem with dogs getting into "animal based" fertilizers that contain things like blood meal, bone meal, fish meal, etc... so I would be concerned about chickens trying to eat those also. So in yards where that might happen, I use a slow-release fertilizer (like osmocote) that is mixed into the soil. You can also use liquids at planting time, or spray on the plant for a "pick me up". Do a "test" with any fertilizer if you are still concerned--see how they react to it.