"Search on" versus "search for"
It's search for. In fact, this is the first time I see search on at all. And indeed, as a quick-and-dirty check, the Corpus of Contemporary American English has a measly 217 cites for "search on" (as opposed to 10.220 for "search for"). The British National Corpus has 39 and 2028 cites, respectively.
You're searching for something on Google, or you're googling something.
When regarding search in general, it is not uncommon to 'search on (field or criteria) to find (desired result)'
Search on the 'Sent On' field to find email that was sent last March.