The missing data issue is ubiquitous in health studies. Variable selection in the presence of both missing covariates and outcomes is an important statistical research topic but has been less studied. Existing literature focuses on parametric regression techniques that provide direct parameter estimates of the regression model. Flexible nonparametric machine learning methods considerably mitigate the reliance on the parametric assumptions, but do not provide as naturally defined variable importance measure as the covariate effect native to parametric models. We investigate a general variable selection approach when both the covariates and outcomes can be missing at random and have general missing data patterns. This approach exploits the flexibility of machine learning modeling techniques and bootstrap imputation, which is amenable to nonparametric methods in which the covariate effects are not directly available. We conduct expansive simulations investigating the practical operating characteristics of the proposed variable selection approach, when combined with four tree-based machine learning methods, XGBoost, Random Forests, Bayesian Additive Regression Trees (BART) and Conditional Random Forests, and two commonly used parametric methods, lasso and backward stepwise selection. Numeric results suggest that when combined with bootstrap imputation, XGBoost and BART have the overall best variable selection performance with respect to the $F_1$ score and Type I error across various settings. In general, there is no significant difference in the variable selection performance due to imputation methods. We further demonstrate the methods via a case study of risk factors for 3-year incidence of metabolic syndrome with data from the Study of Women's Health Across the Nation.