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 $\Bbb K$.

For simplicity, let's take $\Bbb K = \Bbb R$.

By the bundle classification theorem, your question amounts to understanding whether the inclusion map

$$ G_k(\Bbb R^N) \to \underset j{\text{colim }} \, G_{k+j}(\Bbb R^{N+j}) = BO $$ is null homotopic.

First consider the inclusion $$ i: G_k(\Bbb R^N) \to G_k(\Bbb R^\infty) = BO_k \, . $$

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 in degrees $p \le N-k$.

On the other hand, also by Milnor and Stasheff, the restriction homomorphism $$ H^p(BO;\Bbb Z_2) \to H^p(BO_k;\Bbb Z_2) $$ is an isomorphism in degrees $p \le k$. It follows that the homomorphism $$ H^p(BO;\Bbb Z_2) \to H^p(G_k(\Bbb R^N);\Bbb Z_2) $$ is not trivial for all $p$ such that $0 < p \le \min(N-k,k)$. In particular, this is true for some $p >0$ whenever $0 < k < N$.

So the answer to your question is no when $0 < k< N$.

A similar argument works for the other $\Bbb K$.