The generalized normal (GN) distribution has recently attracted enough attention within the research community because of its ability to model additive noise and interference. In this paper, the Type II and VII Pearson distributions as well as the so-called normal-gamma distribution are proposed to approximate the distribution of the sum of GN random variables. The proposed probability distributions can exploit not only the mean and variance but also higher order statistical characteristics, i.e., the kurtosis, in order to accurately approximate the GN sum distribution. Using the moment matching method, simple moment-based estimators for the parameters of these distributions are also presented. It is shown that the proposed approximations yield accurate results over a wide range of distribution parameters and numbers of summands. The proposed approximation is further applied to analyze the error performance of diversity receivers with binary modulation and equal gain combining in the presence of generalized fading and noise. To this end, the characteristic function based approach is employed to transform the average error-rate integral into the frequency domain using the Gil-Pelaez inversion theorem. Based on this approach, the performance of diversity receivers with equal gain combining operating in the presence of double Rayleigh or double Nakagami-m fading environments, which are regarded as appropriate choices to model vehicle-to-vehicle fading channels, is further assessed. Numerically evaluated results accompanied with Monte Carlo simulations are presented to validate the underlying mathematical analysis.