We present experiments on the behavior of reaction fronts in extended, vortexdominated flows in the presence of an imposed wind. We use the ferroin-catalyzed, excitable Belousov-Zhabotinsky chemical reaction, which produces pulse-like reaction fronts. Two time-independent flows are studied: an ordered (square) array of vortices and a spatially disordered flow. The flows are generated with a magnetohydrodynamic forcing technique, with a pattern of magnets underneath the fluid cell. The magnets are mounted on a translation stage which moves with a constant speed V d under the fluid, resulting in motion of the vortices within the flow. In a reference frame moving with magnets, the flow is equivalent to one with stationary vortices and a uniform wind with speed W = V d. For a wide range of wind speeds, reaction fronts pin to the vortices (in a co-moving reference frame), propagating neither forward against the wind nor being blown backward. We analyze this pinning phenomenon and the resulting front shapes using a burning invariant manifold (BIM) formalism. The BIMs are one-way barriers to reaction fronts in the advection-reaction-diffusion process. Pinning occurs when several BIMs overlap to form a complete barrier that spans the width of the system. In that case, the shape of the front is determined by the shape of the BIMs. For the ordered array flow, we predict the locations of the BIMs numerically using a simplified model of the velocity field for the ordered vortex array and compare the BIM shapes to the pinned reaction fronts. We also explore transient behavior of the fronts (before reaching their steady state) to highlight the one-way nature of the BIMs.