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 The Tractor to search for such offset AGNs in a sample of 5493 optically variable AGNs detected with the Zwicky Transient Facility (ZTF). We present the discovery of nine AGNs that may be spatially offset from their host galaxies and are candidates for recoiling SMBHs. Five of these offset AGNs exhibit double-peaked broad Balmer lines, which may have arisen 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 AGNs overall. In our sample of variable AGNs we also identified 52 merging galaxies, including a new spectroscopically confirmed dual AGN. Finally, we detected the dramatic rebrightening of SDSS 1133, a previously discovered variable object and recoiling SMBH candidate, in ZTF. The flare was accompanied by the reemergence of strong P Cygni line features, indicating that SDSS 1133 may be an outbursting luminous blue variable star.