Of course the DOJ/FBI won't comply, just as they originally didn't when Nunes made his requests and ultimately subpoenas for the information he wanted.
Eventually Nunes had to threaten the DOJ with Contempt of Congress, which finally got Rosenstein/Wray to budge, but only after they made their final plea to Ryan, who surprisingly held strong.
Since Johnson is in the Senate, it could play out differently, and may involve Mitch McConnell. Therefore Johnson's request is likely just the first volley in what may be a protracted battle, which is sure to include more "lost" messages, but at least it appears he's taking them head on.
Deadline to hand over the info is Feb 14. From the article, some of the names are below :
Johnson requested the aforementioned communications from Peter Strzok and Lisa Page — the FBI agents who at one point exchanged anti-Trump sentiments before being assigned to special counsel Robert Mueller’s probe — as well as Comey and McCabe.
He also names outgoing FBI chief of staff James Rybicki, acting FBI Deputy Director David Bowdich and John Giacalone, the lead FBI agent on the probe into Clinton’s emails.