To download the vSphere Client for ESXi 5.3 or 5.5, you'll need to visit VMware's official site or search for the exact version in their archives, as those older clients aren't readily available anymore.
The vSphere Client is an application that enables management of a vSphere installation. The vSphere Client provides an administrator with access to the key functions of vSphere without the need to access a vSphere server directly.