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Depends on how you define tornado or hurricane. The sun doesn't really have rotating weather systems that could form such structures. In fact, Earth is fairly unique in how our axial tilt affects our weather.
But the sun certainly has its own kind of storms: https://en.wikipedia.org/wiki/Solar_phenomena
Also, keep in mind, the sun is very, very big compared to Earth. Any of those phenomena look like little bits on the sun, but they're all several times larger than the entire planet Earth.
The question is:
Can a star have a vortex in its outer layers?
I’d find it surprising if it never happened.
The "without water" is a rather dramatic leap. What is a hurricane without water? Basically it doesn't exist. A hurricane critically requires that water evaporates to fuel the storm and later precipitates at its destination, meaning it would be impossible if it weren't for the fact that our atmosphere is near the triple point of water. No such material exists on the sun, everything is a plasma, so there is no evaporation or condensation.
The one thing in a hurricane you can get on the sun is convection, but with a big twist. We get convection in hurricanes in large cells of air that define the storm structure, and those cells stop at the surface, then go up to a max height. On the sun, you don't have a max height (see my last paragraph on water), and you don't have a surface, so you can get convection that travels for extraordinarily long distances. It's a very efficient way to get heat out, so convection also dampens future convection, which can make the whole system daily chaotic, which is what dominates the overall surface structure of the sun. Each bright spot is a single convection zone, and it's falling back down on the dark edges.
