We introduce a finite element method to compute equilibrium configurations of fluid membranes, identified as stationary points of a curvature-dependent bending energy functional under certain geometric constraints. The reparameterization symmetries in the problem pose a challenge in designing parametric finite element methods, and existing methods commonly resort to Lagrange multipliers or penalty parameters. In contrast, we exploit these symmetries by representing solution surfaces as normal offsets of given reference surfaces and entirely bypass the need for artificial constraints. We then resort to a Galerkin finite element method to compute discrete approximations of the normal offset coordinate.