The following is from Lecture notes of Professor Farrell. A hopefully working link is here(page 95):
https://www.dropbox.com/s/80n4wd6xctpe6yr/Characteristic%20Classes%20%28Sparkie%20%E7%9A%84%E5%86%B2%E7%AA%81%E5%89%AF%E6%9C%AC%202014-02-19%29.pdf
We now reach proposition 7. Let $E$ be a complex vector bundle, then the mod 2 reduction of the total Chern class of $E$ is the total Stiefel-Whitney class of $E$. Since there is no Chern classes in odd dimensions, we have there is no Stiefel-Whitney classes in odd dimension as well.
$\textbf{Proof}$
Suppose $E$ is a line bundle. Then we know $w_{0}(L)=1, w_{1}(L)=0,w_{2}(L)=e_{\mathbb{Z}_{2}}(L)=\phi(e(L))$. On the other hand we know $c_{0}(L)=1, c_{1}(L)=e(L)$. So this verified it for line bundles.
For sum of line bundles we have
$$
E=\oplus^{n}_{i=1}L_{i}
$$
So the total Chern class is
$$
c(E)=c(L_{1}) \cup \cdots \cup c(L_{n})\mapsto_{\phi}\omega(E)=\omega(L_{1}) \cup \cdots \cup \omega(L_{n})
$$
We are now going to use splitting principle. But there is a subtle point. We now discuss it.
By the splitting principle there exist
$$
f:\mathcal{B}\rightarrow B
$$
such that
$$
f^{*}(E)=\oplus_{i=1}^{n}L_{i}
$$ and
$$
f^{*}:H^{*}(B,R)\rightarrow H^{*}(\mathcal{B},R)
$$
is monic.
Therefore by naturality and the fact $f^{*}$ is monic with respect to $\mathbb{Z}_{2}$ coefficients.
$$
\phi(f^{*}(c(E)))=f^{*}(\phi(c(E)))=f^{*}(\omega(E))
$$
If $B$ is a CW cplx there is an isomorphism of abelian groups:
$$ (\{\text{iso classes of line bundles}\}, \otimes) \stackrel {c_1} \to (H^2(B),+)$$
Hence the second cohomology of your space classifies line bundles. The question translates to the question about triviality of your bundle. There are quite some methods to check that (especially for line bundles).
Best Answer
Yes. You may use the splitting principle and assume that the vector bundle is a direct sum of $k$ line bundles, and note that the first Chern class of direct sum of line bundles is the sum of first Chern classes of the line bundles, which in turn is the Chern class of the tensor product of those bundles, which is isomorphic the top wedge power of the original vector bundle.