Found the answer in a paper by Ziegler & Richter-Gebert, see Theorem 4.1. Basically, they show that if two oriented matroids $\mathcal{M}_1$ $\mathcal{M}_2$ of ranks $r$ and $r-1$ respectively satisfy $\mathcal{L}(\mathcal{M}_2)\subset \mathcal{L}(\mathcal{M}_1)$ then there exists a (essentially unique) oriented matroid $\mathcal{M}$ with $\mathcal{M}\setminus e=\mathcal{M}_1$ and $\mathcal{M}/e=\mathcal{M}_2$. So the necessary and sufficient condition is the inclusion of covectors.

Found the answer in a paper by Ziegler & Richter-Gebert, see Theorem 4.1. Basically, they show that if two oriented matroids $\mathcal{M}_1$, $\mathcal{M}_2$ of ranks $r$ and $r-1$ respectively satisfy $\mathcal{L}(\mathcal{M}_2)\subset \mathcal{L}(\mathcal{M}_1)$ then there exists a (essentially unique) oriented matroid $\mathcal{M}$ with $\mathcal{M}\setminus e=\mathcal{M}_1$ and $\mathcal{M}/e=\mathcal{M}_2$. So the necessary and sufficient condition is the inclusion of covectors.