The collection of Christoffel symbols $\Gamma_{ij}^k$ of a connection (or of a metric) on a smooth manifold $M$ is not the collection of components of a tensor field in some local chart, i.e. they don't correspond to a section of a tensor bundle over $M$.
Is there a vector bundle, naturally associated to $M$, of which the collection of Christoffel symbols represents a section?
No, the Christoffel symbols are not the components of a section of a natural vector bundle over $M$. Rather, they are the components of a section of a natural affine bundle over $M$, namely the connection bundle $C(M)$, which has the bundle $TM\otimes T^\ast M\otimes T^\ast M$ as its associated vector bundle.
Connections on a vector bundle $E\to M$ are sections of an affine bundle associated to $E$.
Namely there is a vector bundle $J^1E$ of $1$-jets of sections of $E$, and an exact sequence of bundles $$0\to T^*M\otimes E\to J^1E \overset{p}\to E\to 0$$ where the map $p$ is the evaluation ("$0$-jet").
Then a connection is a section of the affine bundle of sections (sic) of $p$, namely the $s\in Hom(E,J^1E)$ such that $p\circ s=id_E$. The associated vector bundle is $Hom(E,T^*M\otimes E)\simeq T^*M\otimes End(E)$, where one can view the Christoffel symbols (if $E=TM$) as living : once (local) a trivialisation is chosen there is an associated "trivial" connexion, and any other connection differs from it by a section of this vector bundle.
