Tensor product vector bundle - bijection from vector valued functions to sections defined by local sections Let $G_i\to P_i\to M$ be principal fiber bundles with representations $\rho_i\colon G_i\to\mathrm{GL}(V_i)$ and associated vector bundles $E_i\to M$. Given local sections $s_i\colon U\to P_i$, I expect that a bijection
\begin{equation}
C^\infty(U,V_1\otimes\cdots\otimes V_n)\to\Gamma(U,E_1\otimes\cdots\otimes E_n)
\end{equation}
can be constructed.
Here's my guess:
For all $m\in U\subset M$,
\begin{align}
V_i&\to (E_i)_m\\
v&\mapsto[s_i(m),v]
\end{align}
is a bijection$^1$ and we define
\begin{equation}
\Phi_m\colon V_1\otimes\cdots\otimes V_n\to(E_1)_m\otimes\cdots\otimes(E_n)_m
\end{equation}
to be the unique isomorphism s.t. $\Phi_m(v_1\otimes\cdots\otimes v_n)=[s_1(m),v_1]\otimes\cdots\otimes[s_n(m),v_n]$.
The isomorphism
\begin{equation}
\Phi\colon C^\infty(U,V_1\otimes\cdots\otimes V_n)\to\Gamma(U,E_1\otimes\cdots\otimes E_n)
\end{equation}
is then defined by
\begin{equation}
(\Phi(f))(m)=(\Phi_m\circ f)(m).
\end{equation}

$^1$This follows from the definition of the equivalence classes and the fact that $G\ni g\to pg\in P$ is a bijection for all $p\in M$ (according to the definition of principal fiber bundles).

So, since you asked in the comments above, here is your confirmation.
Maybe I can add, that since you have sections $s_i$ on $U$ in each of your principal bundles, they are all trivializable over that same $U$, consequently each of the associated vector bundles is trivializable over this $U$.
On the other hand, $C^\infty(U,V_1\otimes\cdots\otimes V_n)$ is a section space over $U$ in the trivial bundle $U\times(V_1\otimes\cdots\otimes V_n)$ and applying the inverse of the respective trivialization to each factor should simply give you the isomorphism
$$
C^\infty(U,V_1\otimes\cdots\otimes V_n)\to\Gamma(U,E_1\otimes\cdots\otimes E_n).
$$
The (inverse) trivilization in each factor is then, as you remarked in the footnote, given by $v\mapsto[s_i(m),v]$.
