You can not configure some other company's email server to hand off emails unless you have an actual account there. Even if you do have an account monit isn't the best program to deal with submitting email. What I would suggest is to just install a local MTA to listen on 127.0.0.1 and then configure monit like this:
set mailserver 127.0.0.1
In that case monit will hand off the email delivery to an actual MTA who can then take care of sending it out, the MTA is perfectly capable of dealing with the remote server being unavailable, unlike monit (since it's not made for that).
How you set up and configure the MTA is out of the scope of this question, however if you want it to send mail directly it would be best to have a static IP, rDNS and mx records. Or you can use a remote smarthost/gateway.
Edit: brief explanation how to install postfix
You will now have configured postfix to send and receive email to and from the internet. Now when you configure monit as described above you will be able to send out email.
Important, in order to improve deliverability you want to make sure that your IP address has a reverse DNS record that resolves back to your domain.
For example, if your domain is example.org and your server is monit.example.org then it should resolve something like this:
monit.example.org has address 188.8.131.52
10.43.0.192.in-addr.arpa domain name pointer monit.example.org
Although it could resolve to a different hostname, just as long as the domain is the same. This is because many email servers will check whether you have a valid rDNS. You can request your ISP (amazon in this case) to change the rDNS for you.