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 $\Bbb K$.