  Composed of ultralight bosons, fuzzy dark matter provides an intriguing
solution to challenges that the standard cold dark matter model encounters on
sub-galactic scales. The ultralight dark matter with mass $m\sim10^{-23}
\rm{eV}$ will induce a periodic oscillation in gravitational potentials with a
frequency in the nanohertz band, leading to observable effects in the arrival
times of radio pulses from pulsars. Unlike scalar dark matter, pulsar timing
signals induced by the vector dark matter are dependent on the oscillation
direction of the vector fields. In this work, we search for ultralight vector
dark matter in the mass range of $[2\times 10^{-24}, 2\times
10^{-22}]{\rm{eV}}$ through its gravitational effect in the Parkes Pulsar
Timing Array (PPTA) second data release. Since no statistically significant
detection is made, we place $95\%$ upper limits on the local dark matter
density as $\rho_{\rm{\tiny{VF}}} \lesssim 5{\rm{GeV/cm^{3}}}$ for $m\lesssim
10^{-23}{\rm{eV}}$. As no preferred direction is found for the vector dark
matter, these constraints are comparable to those given by the scalar dark
matter search with an earlier 12-year data set of PPTA.
