Not every website has an Opensearch compliant XML file implemented into the Website header, so you are not always getting a notification "Web Search" available.
WebOS only seems to support the official OpenSearch way of adding through the html-header XML file and here is a website which generates such an file on the fly:
Generate Search Plugins
Example: https://developer.palm.com/ is not offering a OpenSearch Engine
So you have to do a search on that site for the word TEST (all uppercase) - this would lead to an search result page:
This link has to be pasted into the Search URL: -field on the searchplugins.net site
Search plugin title: goes with the name you want to see in your Just Type Websearch
You can have the icon for the search match the favicon by letting it autodetect.
After hitting Create Plugin the generator creates a new page and if you wait some seconds, a notification will pop up the a new websearch is detected. Now you can choose this in Just Type Preferences and add it to your "Launch and Search"