Select a scion with some dormant buds from a healthy, mature citrus tree that you like. The resulting graft will be a clone of this plant, so choose carefully. Look for an 6 to 8-inch-long stem, less than 1/4 inch in diameter, on a branch that's under a year old. At least a portion of the buds must not be sprouting, and the stem should be stiff enough that the tissue isn't crushed when you cut it. Do this before buds begin to sprout, which is about three or four weeks before the host tree will be ready to accept a graft.
Use a clean, sharp knife to cut the chosen stem from the orange tree. Cut any leaves off. Wrap the scion in a wet paper towel and seal it up in a plastic bag. Store it in your refrigerator crisper until you are ready to bud the orange tree. The scion will remain viable for up to three months. Don't allow it to freeze.
Find a healthy, mature branch, or rootstock, on your orange tree. It should be between 1/2 and 3/4 inch in diameter and receive partial sun daily. A southern or western exposure is best. Pick a readily accessible spot on the limb that will be easy for you to work with. Many gardeners like for their budded material to grow from near the base of the tree, and select locations as low as 6 inches above the ground.
Make a 1-inch long vertical cut through the bark on the top side of the limb. Make a horizontal cut across the branch at the top end of the vertical cut, forming an inverted "T."
Lift gently the flaps of bark formed by the "T," with your fingernail or the tip of a knife, to create a loose pocket. This is where you will tuck your bud. Take your time and work carefully to avoid tearing the flaps. If the bark is not pliable enough to pry easily from the wood, the tree isn't yet ready for budding. Wait and week and try again.
Choose a plump, healthy, unsprouted bud on your scion. Make a tiny cut through the bark and into the wood about 1/2 to 3/4 inch from the bud. Make a second cut perpendicular to the first, about 1/2 to 3/4 inch away from the other side of the bud. Hold your knife at a right angle and slice into the wood under the bud, from one perpendicular cut to the other. Remove the thin sliver of bark and wood from the stick with the bud intact. Don't touch the cut side of the sliver with your fingers to avoid contamination.
Wrap the rest of the scion up in a wet paper towel and seal it back up in the plastic bag. Return it to the refrigerator to hold in reserve to try again should your budding efforts fail.
Slip the edges of the scion bud under the "T" flaps and snugly into the pocket of your host tree branch. This forms a new bud union.
Cover the bud with polyethylene tape. Wrap the branch tightly beginning about 2 inches away from the bud's one side and ending about 2 inches from its other side. The living tissues of the bud and the host tree should soon graft and produce new growth.
Remove the tape two or three weeks later. If the bud is green, it's alive and the graft has been successful. Cut all new emerging shoots within 1 inch of the new bud union from the branch as they appear for the rest of the season. If your bud scion isn't green, try again.