For simplicity, let's take $\Bbb K = \Bbb R$.
By the bundle classification theorem, your question amounts to understanding whether the inclusion map $$ i: G_k(\Bbb R^N) \to G_k(\Bbb R^\infty) = BO_k . $$ is null homotopic.
According to Milnor and Stasheff (page 81), the restriction homomorphism $$ i^* : H^p(BO_k) = H^p(G_k(\Bbb R^\infty)) \to H^p(G_k(\Bbb R^N)) $$ (with any coefficients) is an isomorphism in degrees $p < N-k$. Since $H^p(BO_k;\Bbb Z_2)$ is a polynomial algrbra on the Stiefel-Whitney casses $w_1,\dots,w_k$, it follows that $i^*$ is not trivial for $0 < k < N$.
So the answer to your question is no when $0 < k< N$.
A similar argument works for the other $\Bbb K$.