I would suggest two reasons to think they are referring to simply the Scriptures.
First, there is only one other use of the word search (ἐραυνάω) in the Fourth Gospel, which is in John 5:39. There Jesus says:
You search the Scriptures because you think that in them you have eternal life; and it is they that bear witness about me, yet you refuse to come to me that you may have life.
Since there are no other uses in John's gospel, this inclines me to believe they intend Nicodemus to search the Scriptures.
There are few other uses of the same word in the New Testament as a whole. Romans 8:27, 1 Cor. 2:19, and Rev. 2:23 all deal with the Spirit searching minds/hearts. The only other verse similar at all is 1 Peter 1:10-11, which is difficult in its own right. There the prophets searching intently with great care, inquiring as to the time and person of the Christ. What they are searching there is unclear and may also have to do with the Spirit and so is of little help in our context.
Second, just before this verse in chapter 7, the people are discussing the same thing:
Others said, “This is the Christ.” But some said, “Is the Christ to come from Galilee? Has not the Scripture said that the Christ comes from the offspring of David, and comes from Bethlehem, the village where David was?”
Since the discussion among the crowds regarding the Christ's hometown revolves around the Scripture, there is no reason to think John means anything different here in verse 52. The leaders are simply (and with great irony) putting their weight behind the interpretation that no prophet arises from Galilee, but that the Christ is from Bethlehem.