Not to knit-pick, but it is not believed reptiles and in particular tortoises store D3 in fat cells. Tortoises do not store much energy and have as much need for fat cells as mammals. They store energy reserves in the form of glycogen in the bloodstream and in cells. That is why tortoise do not get fat. There is also no room for fat in a confined tortoise shell. D3 is created in the skin, then carried by the bloodstream to the liver. There it is converted to calcediol. This is the actual substance they check for in a blood sample to check vitamin D levels. It has a life of a few weeks in the bloodstream. Some of this is then carried in the bloodstream to the kidneys and is converted to an active hormone calcetriol, and is the primary agent needed for proper calcium metabolism. This is carried in the bloodstream. It actually governs the uptake of calcium from the gut, or if the diet is very low in calcium, uptake from bones to compensate. Calcediol is also being shown to be a valuable compound for the vital function of many organs.The answer is yes, but to a degree. They wouldn't be getting a whole lot of D3 on a cloudy rainy day while hidden in the bushes, but there are plenty of sunny days even in monsoon season. Our tortoises do not need hours of strong UV every day. They only really need 20-30 minutes once or twice a week. And they can go weeks without any UV because they store it in their fat cells during the times they are producing it.
On a side note - also a misconception many seem to have is tortoises that hibernate (brumate) need to store up fat for the winter. And you must watch to be sure the tortoise does not loose too much weight during hibernation meaning it is using fat stores. When in fact tortoises do not store fat as an energy reserve, as it is glycogen they store and use as energy reserves. During hibernation (brumation) a tortoise should virtually loose no weight at all except a small amount due to water loss. The small amount of energy they need during hibernation is provided by the glycogen stores. In aquatic turtles, the glycogen is also serving as a bit of an antifreeze as an added benefit!