  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.
