This is really a comment try to make clear the point Tilman was trying to make
but it is too long.
A K3 surface has trivial canonical bundle (after all that and simple connectivity is the definition) and hence the bundle
of self dual two forms is trivial (since on a complex surface
we have $\Lambda^+= \Lambda^{2,0} \oplus R \omega$, $\omega$ being the Kahler form) that is to say K3 surfaces admit hyperkahler metrics. Two forms act on vector fields on a four-manifold via contraction then duality under this actions self-dual forms act like imaginary quaternions.  Thus
taking a orthonormal basis of self-dual forms $\omega_1,\omega_2,\omega_3$ and your vector field $X$ you
get a framing (not stable) away from the zeros of $X$,
by looking at $(X,(\iota_X \omega_1)^*,(\iota_X \omega_2)^*,(\iota_X \omega_3)^*)$.
Then arrange that the vector field is point out around each little 3-sphere surrounding a zero
then you see that the induced framing of each little sphere is the Lie-group framing.
I believe this observation is due to Atiyah.