Production of the $Υ(1S)$ meson in association with a vector boson is a rare process in the standard model with a cross section predicted to be below the sensitivity of the Tevatron. Observation of this process could signify contributions not described by the standard model or reveal limitations with the current nonrelativistic quantum-chromodynamic models used to calculate the cross section. We perform a search for this process using the full Run II data set collected by the CDF II detector corresponding to an integrated luminosity of $9.4 fb^-1$. The search considers the $\Upsilon \to \mu\mu$ decay and the decay of the $W$ and $Z$ bosons into muons and electrons. In these purely leptonic decay channels, we observe one ΥW candidate with an expected background of $1.2 \pm 0.5$ events, and one $ΥZ$ candidate with an expected background of $0.1 \pm 0.1$ events. Both observations are consistent with the predicted background contributions. The resulting upper limits on the cross section for $Y + W/Z$ production are the most sensitive reported from a single experiment and place restrictions on potential contributions from non-standard-model physics.