No. A subdivision that can be lifted to a convex cap is called regular (or coherent, or weighted Delaunay). Here is an example of a non-regular subdivision:
For more on this, I recommend the book "Triangulations" by De Loera, Santos, Rambau.
A subdivision can be lifted to a (non-necessarily convex) polyhedron if and only if it admits an equilibrium stress. This is an assignment of real numbers to the edges such that the sum of the forces acting at every vertex is zero. Convex polyhedral lifts correspond to stresses that are positive on the interior edges and negative on the boundary edges. The keywords here are the Maxwell-Cremona correspondence.
To find a stress, one has to solve a system of linear equations. To determine whether there is a positive stress, one has to check whether the solution space intersects the interior of a polyhedral cone.