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).