Probiotics are the same in the dragon world as they are in our world. I have only used probiotics when Smaug was on antibiotics. If your dragon is known to become constipated, meaning they do not poop consistently such as once a day or once every other day, then probiotics may be necessary once a week.
If you have a healthy eater, meaning it eats its bugs and greens, then there is no need to supplement with probiotics.
What other supplements might I need?
Another supplement that I have used often is Critical care. I use Oxbow Critical Care as a food supplement when Smuag needs to either gain weight or if he stopped eating. Oxbow has carnivore and herbavore Critical cares. When smaug was not eating anything, I mixed the two together, but when he needed to just gain weight, I used only the carnivore care.
When using these supplements, you will need a syringe to feed it to your dragon.
Just to be clear, I do not recommend any of these unless your dragon is sick or is not eating! The issue with feeding a healthy dragon these supplements is the chance that they will become obese. If a dragon is overweight then it is most likely going to get fatty liver disease or other illnesses due to the weight gain, so this should only be used if the dragon is not eating or is sick. If your dragon is sick you should see the vet and ask if you should be using critical care.
It should also be said that calcium and vitamin D should be given regularly, these are not supplements they are necessary for the Dragons health. If the dragon is lacking in calcium or vitamin D then it will develop metabolic bone disease and have issues when it is older.