what does it mean? Is it referring to "googling" a word or phrase? Is it more like asking for a book in the library that you don't find on the shelf? Would you be looking for your companion who didn't come back to the tourist bus?
Would "request" be a possible translation?
Not really, that would have been "θέλω να ζητήσω κάτι" or "θα ήθελα να κάνω μια παράκληση" (formal).
Is do a search correct English
Yes, you can say "to do a search".
well... you would normally say "make a search" in English.
I would recommend this as the default.