You do not need the separability of $B$ to define $EF$ for a random vector $F\colon\Omega\to B$; however, you need to assume that $F$ is strongly measurable, in the sense that there is a sequence of finitely-valued random vectors $F_n$ in $B$ such that $\|F_n(\omega)-F(\omega)\|\to0$ for almost all $\omega\in\Omega$. 

By Bochner's theorem, if $F$ is strongly measurable and $E\|F\|<\infty$, then $F$ is Bochner-integrable, in the sense that for some sequence of finitely-valued random vectors $F_n$ in $B$ we have $\|F_n(\omega)-F(\omega)\|\to0$ for almost all $\omega$ and $E\|F_n-F\|\to0$; then $EF:=\lim_n EF_n$, with naturally defined $EF_n$.