yes, that is possible. You can run any scripted code on the remote server by giving that command list to the ssh client, something along the lines
for i in $hostlist ; do
ssh $i “my cmd1; mycmd2;…”
In order not to have to type in each remote’s password, setting up an appropriate public key environment (“authorized_keys”) would be really helpful. And of course you can decide which remote user to contact etc.
Just for completeness: In case you’d be starting remote jobs as background tasks, it’s advisable to locally detach stdin (“ssh -n”) and to properly redirect the output of the remote process at the remote machine. A “hanging” ssh session (your remote script has completed, but the session will not terminate) is a typical indicator for such a situation.
As you have not indicated your scripting skills, I assume your question was actually only concerning the “ssh” remote invocation part.
And like Malcolm already pointed out: If that’s some regular task, in terms of continuous monitoring, there are far better tools to do this, Nagios being near or at the top of such a list.