No. Consider the case of an ellipsoid with three distinct axes, and remove the four umbilic points. Then you cannot find such vector fields on a (punctured) neighborhood of the deleted umbilics. Have a look at this reference and try drawing the vector field on such a punctured neighborhood, and you'll see why.

No. Consider the case of an ellipsoid with three distinct axes, and remove the four umbilic points. Then you cannot find such vector fields on a (punctured) neighborhood of the deleted umbilics. Have a look at this reference on umbilics and try drawing the vector field on such a punctured neighborhood, and you'll see why.