Generate connected induced subgraphs as the satisfying assignments to a SAT instance
It's not possible if the variables are restricted to be only the given ones for the graph vertices. Consider a graph $G$ that has $2n+2$ vertices: $x$, $u_i$, $v_i$, and $y$ for $1\le i\le n$, formed by the union of $n$ edge-disjoint paths $x$--$u_i$--$v_i$--$y$. Also consider only the subgraphs that include both $x$ and $y$: if the whole graph has a small CNF formula for connected subgraphs, then so does this subset of subgraphs, obtained by setting the variables for $x$ and $y$ to true and then simplifying the formula.

The connected subgraphs that include both $x$ and $y$ do have a very simple $2$-DNF formula:
$$\bigvee\limits_{i=1}^n(u_i\wedge v_i).$$
However, when you convert this to its (unique) CNF form, it has $2^n$ terms (one for each way of excluding either $u_i$ or $v_i$ from the subgraph, for each $i$). So no polynomial-size CNF formula exists.

$G$ is not a grid graph but the same construction can easily be embedded into a grid by expanding $x$ and $y$ into larger connected subsets of grid points. As above, the whole graph can't have a small CNF formula, because if it did then you'd also have a small CNF formula for the connected subgraphs that include all the grid points representing $x$ and $y$, but we can write down the formula for this subset of the subgraphs and it does not have polynomial size.