The double cover of the group $SO(n)$ is the spin group $\operatorname{Spin}(n)$.
Do any of the other groups $SU(n)$, $\operatorname{Sp}(n)$, $G_2$, $F_4$, $E_6$, $E_7$, $E_8$ have double covers?
If so what are they?
How do we prove it one way or the other?
Since each of these groups can be written as a subgroup of $SO(m)$ for some $m$, then this suggests to me there may be double covers that are the subgroup of $\operatorname{Spin}(m)$. Is this logic correct?
$SU(n)$ is simply connected for $n \ge 2$ and $Sp(n)$ is simply connected for $n \ge 1$, so they don't have nontrivial connected covers (you can take their preimages in the spin group with respect to some embedding in an orthogonal group but they won't be connected).
For the others, a priori it depends on what group you mean. You've given names of Lie algebras and those Lie algebras correspond to many Lie groups. According to Wikipedia, the compact real form of $G_2$ is simply connected and has trivial center, so it's the unique connected Lie group with Lie algebra the compact real form of $\mathfrak{g}_2$ and it has no nontrivial connected covers. Wiki also claims the same for $F_4$ and $E_8$ but says that there's a compact real form of $E_7$ with fundamental group $C_2$ and a compact real form of $E_6$ with fundamental group $C_3$. So there's a compact real form of $E_7$ with a nontrivial double cover. I don't know how to verify any of this.