A very interesting statement, in my opinion, that is proved in the famous Serre-Swan paper is the following lemma
Lemma: Let $X$ be a compact topological space and $\xi \rightarrow X$ a vector bundle . Then there exists a trivial vector bundle $\eta$, i.e. $\eta=X\times \mathbb{R}^n$ for some n, and a vector bundle epimorphism $\eta \rightarrow \xi$ over the identity map.
It is an immediate corrolary that any vector bundle over a compact topological space can be completed to a trivial vector bundle, in the sense that it is a direct summand of a trivial vector bundle over X.
It seems to me that the compactness assumption is critical in the proof of this Lemma, however, I was wondering if the statement is also true for vector bundles over arbirtary smooth manifolds.
This is true. In fact, any vector bundle over something homotopy equivalent to a finite dimensional CW complex (for example, over a non compact manifold) is a summand of a trivial bundle.
To see this, note that you just need to prove it for tautological bundle of a Grassmannian, since every vector bundle over such a space is pulled back from the tautological bundle over a Grassmannian. The tautological $k$-plane bundle $E_k \to \mathrm{Gr}_{k} \mathbb{R^n}$ is a summand of the trivial $n$-plane bundle over $Gr_{k} \mathbb{R}^n$, which finishes the proof.
By the way, its not true that vector bundles over any space have a complement, consider the universal line bundle over $\mathbb{RP}^{\infty}$. Since $H^*(\mathbb{RP}^{\infty}, \mathbb{F}_2 )$ is a polynomial ring with generator the first Stieffel-Whitney class of the universal bundle, this bundle doesn't have a complement.