No
Thousand island dressing is usually not vegan.
It's almost always based on mayonnaise, which includes egg yolk. While brands and recipes may vary slightly, other common non-vegan ingredients include Worcestershire sauce and finely chopped hard boiled eggs.
There are many recipes you can follow to create your own vegan-friendly thousand island from home. This one, for example, substitutes traditional mayo with a vegan mayo.
While making your own dressing from home is a great way to control and know exactly what is going into your dressing, buying a store-bought variety can be a bit riskier. These versions can contain a wide array of preservatives, flavorings, and colors - not all of which are guaranteed to be vegan. Unless you've contacted the company or can see the word "vegan" somewhere on the bottle, it may be best to avoid it.
Also see: Is Thousand Island Dressing Bad For You?