The problem of recognizing whether a simplicial face lattice is polytopal is sometimes called the Steinitz problem.
Sturmfels and Bokowski advanced a set of methods in the late 80s to test whether the face lattice of a simplicial sphere was also realizable as a polytope.
The method uses oriented matroids. The problem is NP-hard, so their algorithm requires exponential time in the worst case, but they reported that the algorithm often converged quickly.
In the intervening two decades, I'm sure that newer approaches have been developed. Is there a better method known today? More interestingly, are there any software implementations available that solve this problem -- even using the older approach?