  We study the cross section for vector boson ($W^{\pm}/Z^0/\gamma^*$)
production in polarized nucleon-nucleon collisions for low transverse momentum
of the observed vector boson. For the case where one measures the transverse
momentum and azimuthal angle of the vector bosons, we present the cross
sections and the associated spin asymmetries in terms of transverse momentum
dependent parton distribution functions (TMDs) at tree level within the TMD
factorization formalism. To assess the feasibility of experimental
measurements, we estimate the spin asymmetries for $W^{\pm}/Z^0$ boson
production in polarized proton-proton collisions at the Relativistic Heavy Ion
Collider (RHIC) by using current knowledge of the relevant TMDs. We find that
some of these asymmetries can be sizable if the suppression effect from TMD
evolution is not too strong. The $W$ program at RHIC can, thus, test and
constrain spin theory by providing unique information on the universality
properties of TMDs, TMD evolution, and the nucleon structure. For example, the
single transverse spin asymmetries could be used to probe the well-known Sivers
function $f_{1T}^{\perp q}$, as well as the transversal helicity distribution
$g_{1T}^{q}$ via the parity-violating nature of $W$ production.
