Let $P(M,G)$ be a principal $G$-bundle. A connection on $P(M,G)$ is given by a distribution $\mathcal{H}\subseteq P$ satisfying some properties. Equivalently, it is given by a differential form $\omega:P\rightarrow \Lambda^1_{\mathfrak{g}}T^*P$ satisfying some conditions. This is easy to handle when looking for pullback of connections along morphism of principal bundles.
But, as far as I know, we do not have such set up for case of connection on a vector bundle $\pi: E\rightarrow M$. What I know is, given a section $s:M\rightarrow E$ of $\pi:E\rightarrow M$, we have an associated $1$-form $\nabla_s:M\rightarrow \Lambda^1_ET^*M$. The letter $E$ in $\nabla_s$ indicates that this is a vector bundle valued $1$-form.
I think there is no single differential form at any level, either on $M$ or on $E$ that remembers the connection fully. Is this correct? Or am I misunderstanding something here? One can always think of associated principal $GL(n,\mathbb{R})$-bundle, denoted by $Pr(E)\rightarrow M$ and think of connection $1$-form there. This is not what I am asking for.
Is there a one form (or a higher degree form) on $E$ or $M$ (vector bundle valued is also fine) that remembers the connection on the vector bundle $\pi:E\rightarrow M$?
Suppose I have a morphism of Principal bundles $P(M,G)\rightarrow P’(M’,G’)$ such that the map $G\rightarrow G’$ is an isomorphism, then, given any connection on $P’(M’,G’)$, one can talk about pullback of this connection to get a connection on $P(M,G)$. Reference is Kobayashi and Nomizu, Volume $1$, Proposition $6.2$.
This does not happen (as far as I know) in case of vector bundles. Suppose I have a morphism of vector bundles $(E,\pi, B)\rightarrow (E’,\pi’, B)$, and a connection on $(E’,\pi’, B)$, I do not know if there is any obvious way to construct a connection on $(E,\pi, B)$. All I can do, is to define a connection on pullback of vector bundle along a smooth map. This is what I mean when I say having a 1-form helps to pullback the notion of connection which does not seem to happen in case of connection on vector bundles.
There are (too) many points of view on connections. A connection is not a pointwise object so that there is no hope of representing it as a section of a tensor bundle over $M$. However, both $G$-connections on principal bundles and linear connections on vector bundles are instances of Ehresmann connections which make sense for general smooth fiber bundles and such connections can be described as bundle-valued one-forms on the total space.
For the case of vector bundles, an equivalent definition of a linear connection is a distribution on $TE$ which complements the vertical bundle $VE = \ker{d\pi}$ and satisfies some "symmetry" conditions (to make sure that parallel transport is linear). Instead of describing the horizontal bundle, you can just give a morphism $\omega \colon TE \rightarrow VE$ which is the projection onto the vertical bundle (the kernel will be the horizontal bundle). Such an object is called a "connection form" (see here, although my notation and types are slightly different) and it is naturally a $VE$-valued one-form on $E$. Given a section $s \colon M \rightarrow E$ of $E$, you can pull back $v$ along $s$ to get a one-form $s^{*}(v) \in \Omega^1(M;s^{*}(VE))$ on $M$ and under the identification of $s^{*}(VE) \cong E$, this is just $\nabla s$.