In this paper, we consider the empirical spectral distribution of the sample correlation matrix and investigate its asymptotic behavior under mild assumptions on the data's distribution, when dimension and sample size increase at the same rate. First, we give a characterization for the limiting spectral distribution to follow a Marchenko-Pastur law assuming that the underlying data matrix consists of i.i.d. entries. Subsequently, we provide the limiting spectral distribution of the sample correlation matrix when allowing for a dependence structure within the columns of the data matrix. In contrast to previous works, the fourth moment of the data may be infinite, resulting in a fundamental structural difference. More precisely, the standard argument of approximating the sample correlation matrix by its sample covariance companion breaks down and novel techniques for tackling the challenging dependency structure of the sample correlation matrix are introduced.