Many risk-sensitive applications require well-calibrated prediction sets over multiple, potentially correlated target variables, for which the prediction algorithm may report correlated errors. In this work, we aim to construct the conformal prediction set accounting for the joint correlation structure of the vector-valued non-conformity scores. Drawing from the rich literature on multivariate quantiles and semiparametric statistics, we propose an algorithm to estimate the $1-α$ quantile of the scores, where $α$ is the user-specified miscoverage rate. In particular, we flexibly estimate the joint cumulative distribution function (CDF) of the scores using nonparametric vine copulas and improve the asymptotic efficiency of the quantile estimate using its influence function. The vine decomposition allows our method to scale well to a large number of targets. As well as guaranteeing asymptotically exact coverage, our method yields desired coverage and competitive efficiency on a range of real-world regression problems, including those with missing-at-random labels in the calibration set.