A supermassive black hole (SMBH) ejected from the potential well of its host galaxy via gravitational wave recoil carries important information about the mass ratio and spin alignment of the pre-merger SMBH binary. Such a recoiling SMBH may be detectable as an active galactic nucleus (AGN) broad line region offset by up to 10,kpc from a disturbed host galaxy. We describe a novel methodology using forward modeling with texttt{The Tractor} to search for such offset AGN in a sample of 5493 optically variable AGN detected with the Zwicky Transient Facility (ZTF). We present the discovery of 9 AGN which may be spatially offset from their host galaxies and are candidates for recoiling SMBHs. Five of these offset AGN exhibit double-peaked broad Balmer lines which may arise from unobscured accretion disk emission and four show radio emission indicative of a relativistic jet. The fraction of double-peaked emitters in our spatially offset AGN sample is significantly larger than the 16% double-peaked emitter fraction observed for ZTF AGN overall. In our sample of variable AGN we also identified 52 merging galaxies, including a new spectroscopically confirmed dual AGN. Finally, we detected the dramatic rebrightening of SDSS1133, a previously discovered variable object and recoiling SMBH candidate, in ZTF. The flare was accompanied by the re-emergence of strong P-Cygni line features indicating that it may be an outbursting luminous blue variable star.