Transplanting sunflowers isn't difficult and is similar to transplanting other plants. The main thing to keep in mind is that the stems on young sunflowers are very delicate and may require a bit of care for the first weeks.
Prepare the ground for planting
Dig holes for the plants. Remember that sunflowers are tall and so the roots spread out farther than you might expect. Space the holes far enough apart the plants aren't crowded once they get larger. The holes should be 3 to 4 inches deep with loose soil all around so the roots have something to grab.
Put the sunflower into the hole
Carefully place the sunflower into the middle of the hole, and loosely cover the roots. Don't pack the soil too tightly in the hole.
Sunflowers are hardy and don't need to be fertilized unless your soil is seriously depleted. They don't need great quantities of water, but watering helps the plant overcome transplanting trauma and settle into its new home.
If you are transplanting very small plants, staking them with popsicle sticks or other small wooden or plastic supports is helpful. It isn't necessary in all cases, but strong wind or heavy rain can bend the plants and damage those delicate stems. Attach the plant to the stakes carefully so you don't tear the stems, bruise the leaves or prevent natural growth.