My topology and maths are very rusty, am a software developer these days.

I think that there are both tangent vector fields that don't and some that do. In the two dimnsional case (circle) certainly all do.

In n I intuitively would say that you should be able to have a vector field that does but I am now less confident to think about a proof on my bus rides while I answer here. I tried twice already.

I will try to think about this more, will ping here if I get more