To back up machines in an ESXi Free environment, regular Acronis Agent for Windows/Agent for Linux (as opposed to Acronis Agent for ESX/ESXi) should be installed inside the virtual machines. Agentless backup is not supported.
The reason is that the specific VMware API through which Acronis Agent for ESX/ESXi (virtual appliance) works is the Remote Command Line (RCLI) appliance. RCLI is available in read-only access in the free VMware ESXi. Due to this, Acronis Agent for ESX/ESXi cannot function in the free VMware ESXi.