Is it possible to set up an email address on my non-Plesk shared hosting email server to forward to more than one email address?

Yes, it is possible to add multiple forwards for the same email address. To do so:

First, add one of the forwards as usual.
Then go back to the main menu and click the Forwards link.
Click the Modify button next to the notifications@domain.com forward.
In the Destination field, you can enter an additional forwarding address.
Finally, click Add. You should then see both of them shown in the list at the top.

Repeat this process to add additional forwards as needed.