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 or Acronis Agent for VMware) should be installed inside the virtual machines. Agentless backup is not supported.
The reason is that the VMware vStorage APIs are not installed on the ESX/ESXi server, as this option is not available in the free version of VMware ESXi. Due to this, Acronis Agent for ESX/ESXi cannot function in the free VMware ESXi.
You can check if this feature is installed in Configuration -> Licensed Features on the ESX/ESXi server: