Let $M$ be a closed orientable connected topological manifold of dimension $4k$ with $k > 1$. It is known (David Frank, On the index of a tangent 2-field) that if $M$ has a smooth structure, the necessary and sufficient condition for it to admit two linearly independent vector fields is the following:

The Euler characteristic $\chi(M) = 0$ and the signature $\sigma(M) \equiv 0 \pmod{4}$.

Do the same two conditions characterize whether the tangent microbundle $\tau M$ of $M$ has a rank 2 trivial subbundle?

The vanishing $\chi(M) = 0$ is evidently necessary, for instance by the Lefschetz fixed point theorem; it is also sufficient for $\tau M$ to have a rank one trivial subbundle by Corollary 1.5 in Ronald J. Stern, On topological and piecewise linear vector fields.

Duane Randall has papers that work out the analogous characterization with non-orientable manifolds and manifolds of dimension $4k+1$, $4k+2$, $4k+3$, but they seem to leave the orientable 4k case unsettled in general.
Duane Randall, CAT 2-fields on nonorientable CAT manifolds
Duane Randall, On indices of tangent fields with finite singularities
Duane Randall, On 4-manifolds and span-related numbers for CAT manifolds