Let $A$ be an $\bar{\mathbb{F}}_p$-Algebra of finite type (one might assume $A$ to be reduced). Let $X \subset \mathbb{A}_A^d\backslash \{0\}$ be a closed $A$-subscheme together with a group action of $\mathbb{G}_{m,A}$ induced from the action on $\mathbb{A}_A^d\backslash \{0\}$. Under those assumptions, does there exist a categorical quotient $X/\mathbb{G}_{m,A}$ which is a finite type scheme over $A$?
I think this should work in the case $A= \bar{\mathbb{F}}_p$, as $\mathbb{G}_{m,A}$ is then linearly reductive. But I know too few (references) for more general $A$-group scheme actions.
You can just take this to be the image of $X$ in $\mathbb{P}_A^{d - 1}$.
To express the above comment about "image" in more precise terms, since for any ring $A$ at all the natural map$$\pi: \mathbb{A}_A^d - \{0\} \to \mathbb{P}_A^{d - 1}$$is a $\mathbb{G}_{m, A}$-torsor, a closed subscheme $X$ of $\mathbb{A}_A^d - \{0\}$ is $\mathbb{G}_{m, A}$-stable if and only if the associated ideal sheaf is stable under the descent datum for $\pi$. Hence, descent theory for quasicoherent sheaves provides a unique closed subscheme $Z \subset \mathbb{P}_A^{d - 1}$ over which $X$ is a $\mathbb{G}_{m, A}$-torsor and it is the desired "quotient" of $X$ (over $A$) in all good senses.