In the modern world, we tend to think of anybody famous as being a celebrity, and celebrities should sign autographs and take pictures with their fans. That doesn't apply to the Royal Family. They're not celebrities, and their standard response when asked to sign anything is 'They don't let me do that.' It's not that they don't want to but concerns about forgery of Royal signatures means they're not allowed.
The current Royals have on occasion deviated from this norm, though; possibly because signatures don't carry a much weight as they once did. Prince Charles was seen signing autographs while touring areas of Cornwall affected by a flood in 2010, Prince Harry will take pictures (but not 'selfies' as he hates them!) with people who ask politely, and the Queen famously and deliberately 'photobombed' two hockey players in Glasgow in 2014. Who says they don't have a sense of humor!