On a $G$-principal bundle $P$ it's easy since $G$ has a preferred element, namely $e$ (identity), but there is no such element in arbitrary suitable space $F$ to construct local sections for the associated bundle to $P\times_{\rho} F$ where $\rho:G\longrightarrow \mathrm{Aut}(F)$
I tried:
$$\tilde{s}_{\alpha}(m)=[(s_{\alpha}(m),(f\circ s_{\alpha})(m))]$$
for some $f:P\longrightarrow F$ where $s_{\alpha}$ is a local section of $P$ and $[(p,f)]$ denotes the class of $P\times_{\rho} F$ which $(p,f)$ is the representant, but it is not well defined.