Absolutely it can be done; but I don't know who does your administration work since you're on Remedy on Demand. A business rule can be built that when a critical or high comes in, fire off an email. Might even be possible to do through the NTE notifications forms, not sure on that. (If so wouldn't even need a business rule.)
Is there any other way to do this automation if one doesn't have the admin rights? Something from within the application itself? How to do it through NTE Notification form on Remedy On Demand and where will I find this form in it?
If you have the proper access, you can access the NTE forms such as
NTE:SYS-Define NT Events
and those might work for you; you can get to them with the web client, you don't need Dev Studio.
I recommend reading the ITSM Notification Engine Guide from support to get information on how to use the notification engine and to see if it will work for you.
In my opinion NTE forms will give you limited options to control the flow of emails.
A quick solution to your requirement is to create workflows on the incident form which can trigger emails on submit of a critical/high ticket. You will be able to choose the format of the email and decide the recipients for that email.
For doing this you can contact the OnDemand Customer Support who will have a technical SME get in touch with you and do this for you. (Ofcourse if you have purchased REH - Remote Enhancement Hours). Alternatively you can get in touch with the BMC sales rep to purchase the REH.
Hope this information is useful to you.
Thanks & Regards,