  We report the first measurement of transverse single-spin asymmetries in
$J/\psi$ production from transversely polarized $p+p$ collisions at $\sqrt{s} =
200$ GeV with data taken by the PHENIX experiment in 2006 and 2008. The
measurement was performed over the rapidity ranges $1.2 < |y| < 2.2$ and $ |y|
< 0.35$ for transverse momenta up to 6 GeV/$c$. $J/\psi$ production at RHIC is
dominated by processes involving initial-state gluons, and transverse
single-spin asymmetries of the $J/\psi$ can provide access to gluon dynamics
within the nucleon. Such asymmetries may also shed light on the long-standing
question in QCD of the $J/\psi$ production mechanism. Asymmetries were obtained
as a function of $J/\psi$ transverse momentum and Feynman-$x$, with a value of
$-0.086 \pm 0.026^{\rm stat} \pm 0.003^{\rm syst}$ in the forward region. This
result suggests possible nonzero trigluon correlation functions in transversely
polarized protons and, if well defined in this reaction, a nonzero gluon Sivers
distribution function.
