The Generalized Linear Model (GLM) for the Gamma distribution (glmGamma) is widely used in modeling continuous, non-negative and positive-skewed data, such as insurance claims and survival data. However, model selection for GLM depends on AIC/BIC criteria, which is computationally impractical for even a moderate number of variables. In this paper, we develop variable selection for glmGamma using elastic net regularization (glmGammaNet), for which we provide an algorithm and implementation. The glmGammaNet model is more challening than other more common GLMs as the likelihood function has no global quadratic upper bound, and we develop an efficient accelerated proximal gradient algorithm using a local model. We report simulation study results and discuss the choice of regularization parameter. The method is implemented in the R package glmGammaNet.