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][1] and try drawing the vector field on such a punctured neighborhood, and you'll see why.


  [1]: https://en.wikipedia.org/wiki/Umbilical_point "this"