The phrasal verb "back up" has been combined to make a noun "backup" or "backups". But the verb form still has its suffixes after "back", not after "up". The servers should be "backed up", not "backupped" (and certainly not "backuped").
You might want to change your sentence to something like:
Started searching for servers that should be backed up
In this context, "searching of servers" implies that you are searching the contents of the servers, whereas "searching for servers" implies that you are trying to find the servers.
Either "which" or "that" is okay. "that" seems more natural to my (American) ear.
You can choose whether your logs have the form "<Action> {started / completed}" or "<Verb> <action>". My rewrite uses the second form. The second form implies a sentence like "A script has started searching for servers that should be backed up."
By the way, you can use "done" or "finished" instead of "completed".