Let's say I'm doing some research on something on Google. Do I say I'm searching on or searching for something on Google?
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.