Yes
There have been some doubts in the vegan community on whether Skittles can be considered vegan-friendly. The reason for this controversy was the gelatin content, which was present in the Skittles sold in the US up until 2009. Since then, Skittles have become gelatin-free, and they don't contain any animal-based ingredients.
Hydrogenated Palm Kernel Oil, a derivative of palm oil, is another controversial ingredient used in Skittles. Although palm oil doesn't contain any animal products, it is one of the major causes of deforestation, which leads to a large number of animal deaths each year. Since most members of the vegan community care deeply for this issue, it is good to be aware of it. However, as far as the ingredients are concerned, we can safely conclude that Skittles are vegan.
Also see: Are Skittles Bad For You?