In short: The idempotent completion of the essential image of $AbVar_{/\mathbb{F}_q}$ in $FG_{/\mathbb{F}_q}$ is categorically equivalent to $FG_{/\mathbb{F}_q}$.

In Manin's paper, The Theory of Formal Commutative Groups over Fields of Finite Characteristic, he calls formal groups "algebroid" if they are the completion of algebraic varieties over any field $k$. He proves that every formal group over $\mathbb{F}_q$ is isogenous to a subgroup of an algebroid formal group over $\mathbb{F}_q$. The proof outline is:

1. Every formal group is isomorphic to $\oplus_i G_{n_i,m_i}$ up to isogeny.
2. Every such $G_{n,m}$ (pick any coprime m, n) can be realized in the formal group decomposition associated to an abelian variety.

Notation: $G_{n,m}$ is the formal group whose Dieudonné $W(\mathbb{F}_p)$-module is isomorphic to $W(\mathbb{F}_p)/(F^n - V^m)$.

For step 2, he uses the Jacobian variety $J_a$ associated to the family of curves $y^p - y = x^{p^a-1}$ over $\mathbb{F}_{p^a}$, where $a = 1, 2, 3, ...$. The decomposition of the formal group $\hat{J}_a$ contains all $G_{n, m}$ such that $m+n \mid a(p-1)$. So, for any pair of coprime $(n, m)$ and any $p$, $\hat{J}_{n + m}$ contains a piece isogenous to $G_{n,m}$.

(There is the subtlety that we are working in $FG$ up to isogeny rather than isomorphism here, so this answer only tells us about the category $FG$ up to isogeny.)

A note on the difference in base field from my original question: I asked about $\mathcal{O}_{K_p}$ where $K_p$ is a finite extenstion of $\mathbb{Q}_p$, whereas Manin considers $\mathbb{F}_q$. It seems that since the varieties $A$ are defined over finite extensions of $\mathbb{Q}_q$, if they have good reduction we can extend them (via the Néron model?) to $\mathbb{Z}_q$, and take their special fiber to get to varieties defined over $\mathbb{F}_q$.

In short, I still don't know over a general ring, but for $\mathbb{F}_q$, we have an answer: The idempotent completion of the essential image of $AbVar_{/\mathbb{F}_q}$ in $FG_{/\mathbb{F}_q}$ is categorically equivalent to $FG_{/\mathbb{F}_q}$.

In Manin's paper, The Theory of Formal Commutative Groups over Fields of Finite Characteristic, he calls formal groups "algebroid" if they are the completion of algebraic varieties over any field $k$. He proves that every formal group over $\mathbb{F}_q$ is isogenous to a subgroup of an algebroid formal group over $\mathbb{F}_q$. The proof outline is:

1. Every formal group is isomorphic to $\oplus_i G_{n_i,m_i}$ up to isogeny.
2. Every such $G_{n,m}$ (pick any coprime m, n) can be realized in the formal group decomposition associated to an abelian variety.

Notation: $G_{n,m}$ is the formal group whose Dieudonné $W(\mathbb{F}_p)$-module is isomorphic to $W(\mathbb{F}_p)/(F^n - V^m)$.

For step 2, he uses the Jacobian variety $J_a$ associated to the family of curves $y^p - y = x^{p^a-1}$ over $\mathbb{F}_{p^a}$, where $a = 1, 2, 3, ...$. The decomposition of the formal group $\hat{J}_a$ contains all $G_{n, m}$ such that $m+n \mid a(p-1)$. So, for any pair of coprime $(n, m)$ and any $p$, $\hat{J}_{n + m}$ contains a piece isogenous to $G_{n,m}$.

(There is the subtlety that we are working in $FG$ up to isogeny rather than isomorphism here, so this answer only tells us about the category $FG$ up to isogeny.)

A note on the difference in base field from my original question: I asked about $\mathcal{O}_{K_p}$ where $K_p$ is a finite extenstion of $\mathbb{Q}_p$, whereas Manin considers $\mathbb{F}_q$. It seems that since the varieties $A$ are defined over finite extensions of $\mathbb{Q}_q$, if they have good reduction we can extend them (via the Néron model?) to $\mathbb{Z}_q$, and take their special fiber to get to varieties defined over $\mathbb{F}_q$.