Royal icing contains egg white so isn't vegan friendly. Here are some solutions.
There are two solutions if you want to use a plant based royal icing:
i) buy it ready made. Renshaw's for example do a ready to roll Royal Icing that is Vegan friendly and Kosher (and quite affordable). It's available from Amazon, Iceland and craft shops which have specialist cake icing lines.
ii) make it. Here's the recipe:
For the Royal Icing: 12 Tablespoons Aquafaba (Chickpea Brine) 800g icing sugar (ensure its vegan friendly) 3 teaspoon Glycerine (check the label to ensure the glycerin is made with vegetable fat not animal fat)
Whilst they are cooling, you can make the icing. Beat the Aquafaba with a whisk until stiff. Then add the icing sugar and Glycerine and beat until smooth. Spoon into a piping bag and use according to the recipe instructions.