No, there is no Linux-based appliance for Veeam Monitor. And yes, Veeam Monitor has to monitor the corresponding host 24/7 if you want to receive alarms and react proactively.
What about installing Veeam Monitor on any machine (can be any existing physical host or VM) and target it to collect information from that ESXi free host? Do you have any machines running all the time?
Also please look through our system requirements section in the Release Notes
document for the minimum resources required.