Cloud Radio Access Network (C-RAN) is a prominent architecture for 5G wireless cellular system that is based on the centralization of baseband processing for multiple distributed radio units (RUs) at a control unit (CU). In this work, it is proposed to leverage the C-RAN architecture to enable the implementation of direct localization of the position of mobile devices from the received signals at distributed RUs. With ideal connections between the CU and the RUs, direct localization is known to outperform traditional indirect localization, whereby the location of a source is estimated from intermediary parameters estimated at the RUs. However, in a C-RAN system with capacity limited fronthaul links, the advantage of direct localization may be offset by the distortion caused by the quantization of the received signal at the RUs. In this paper, the performance of direct localization is studied by accounting for the effect of fronthaul quantization with or without dithering. An approximate Maximum Likelihood (ML) localization is developed. Then, the Cramer-Rao Bound (CRB) on the squared position error (SPE) of direct localization with quantized observations is derived. Finally, the performance of indirect localization and direct localization with or without dithering is compared via numerical results.