Let $\mathrm{Gr}_n$ denote the infinite real Grassmannian of $n$-planes in $\mathbb{R}^\infty$. This is a classifying space for real vector bundles, in the sense that (for paracompact $B$) $$ [B, \mathrm{Gr}_n] \cong \mathrm{Vect}_n(B).$$ By more elementary arguments, we have an isomorphism $$ [S^3,\mathrm{GL}(n,\mathbb{R})] \cong \mathrm{Vect}_n(S^4),$$ namely by cutting up $S^4$ as an upper and lower hemisphere and considering their intersection, which is homotopy equivalent to $S^3$.
I was curious whether the second relation could be derived from the first. I asked around for this a bit, and someone told me that $\Omega \mathrm{Gr}_n \simeq \mathrm{GL}(n,\mathbb{R})$ (the loop space), and then the adjunction between suspension and loop space would yield the result. The only trouble I see with this is that this adjunction only exists when using pointed homotopy classes, which does not appear in the above isomorphisms classifying vector bundles. Is there a way to remedy this?
Additionally, if someone has a reference for the fact that $\Omega \mathrm{Gr}_n \simeq \mathrm{GL}(n,\mathbb{R})$, I would appreciate it.
The fact that $\Omega Gr_n \simeq GL(n,\mathbb{R})$ follows from the fact that $Gr_n$ is the classifying space of $GL(n,\mathbb{R})$. In general, $\Omega BG \simeq G$.
Now for your initial question: in order to use this adjunction we need to see that two maps are homotopic, if and only if any basepointed representatives are basepoint homotopic. Recall that isomorphism classes of n-dimensional vector bundles over a basepointed CW complex X correspond to homotopy classes of maps $X \rightarrow Gr_n$. Now if we let the basepoint of $Gr_n$ be $\mathbb{R}^n$ sitting inside $\mathbb{R}^\infty$, then we want to understand basepointed homotopy classes of maps from $X$ to $Gr_n$. These correspond to vector bundles with fiber over $* \in X$ equal to $\mathbb{R}^n$ up to isomorphism of vector bundles that restricts to the identity over $*$.
So given two basepointed vector bundle $V, V'$ and an isomorphism between them, take a nice enough closed trivializing neighborhood $U$ around $*$. Then we can identify our vector bundle in this neighborhood with $U \times \mathbb{R}^n$ so that it is the identity at the basepoint. Then we take a map $U \times \mathbb{R}^n \rightarrow GL(n,\mathbb{R})$ such that it is the the inverse of $V|_* \rightarrow V'|_*$ on $*$ and $0$ on $\partial U \times \mathbb{R}^n$. We can then act on $V'$ via this map to get an isomorphism of vector bundles that restricts to the identity on the basepoint. Hence, basepointed and free homotopy classes of maps into $Gr_n$ coincide.