If you are feeding a perfect diet, no supplements are necessary. In captivity we can not easily reach perfection, therefore vitamins should be given, usually once or twice weekly.
Calcium: Turtles need calcium to build healthy shells and bones. A cuttle fish bone in the tank will be nibbled on by most turtles and supplies extra calcium. Get a new bone when the old one get slimy! There are also calcium supplements specially for reptiles.
Vitamin D3: This is synthesized in the turtle`s body using natural sunlight. A full spectrum reptile light is an acceptable substitute, or a supplement can be fed. There are supplements available that contain Ca/D3. Of course, the best way for your turtle get D3 is to get unfiltered sunlight, so it can synthesize the vitamin. NOTE: OUTDOOR TURTLES SHOULD NOT BE SUPPLEMENTED WITH VITAMIN D3, as they make plenty of their own.
Vitamin A: If lacking, will cause loss of appetite and swollen, runny eyes; A common symptom in turtles not fed properly. This Can be fixed using a varied diet. In severe cases, drops may be recommended by your vet. Feeding too much Vitamin A will cause the skin to peel and eventually come off and leave bare flesh. It looks very ugly, is very painful, and often the turtle will die. Vets used to give turtles vitamin A shots. But since dosage is hard to determine, this should only be used as a last resort. There are some products available now that are formulated for reptiles. Herptivite (by RepCal) is a recommended product.