All I can find online is stuff about search result rankings, which I’m not concerned with at the moment. We’re going to have a cloud-based online database and we want Google search results for something relevant to our site to display our website’s search results for the same query.
For example, if someone types in “lunchbox” to google, there might be a result that says “Amazon search results for ‘lunchbox'” and then you click that and it takes you to amazon, and it’s already searched ‘lunchbox’ for you.
What can we do to enable this capability for our site?
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You can restrict Google’s search results to a specific site with the site: directive.
lunchbox +site:amazon.com
will tell Google to only query “lunchbox” on domains at/under “amazon.com” (that last bit is important: “+site:stackexchange.com” gets you hits on the entire network).
I re-read your question, and am not sure that my first interpretation was correct. It now seems like you’re actually saying “when someone comes from Google, I want to re-run my own search with the same search terms.”
If that’s the case then the answer’s easy: look at the Referer header, and pull the terms out of it.
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