Vilips carded a flawless six-under 65 at breezy Harbour Town on Friday with four birdies and an eagle.
He was tied in 13th place - six shots from the lead - in a group including Davis, who had six birdies and would have matched Vilips' round score but for a bogey at the last.
It is shaping up to be a wide-open weekend at the $US20 million signature event which has a 72-player field and no halfway cut.
The man everyone is chasing is American star Justin Thomas, who salvaged a 69 for a two-shot lead at 12 under - his first 36-hole lead in more than four years.
Two-time major winner Thomas is winless since the 2022 PGA Championship and is weary of the topic, though his game has been trending in the right direction and he is ranked world No.8.
He also knows there's a long way to go on a course that demands precision while allowing a low score.
"The majority of it has just been pressing and trying too hard," Thomas said of his drought. "I almost feel like I'm kind of past the point where fortunately you all have stopped asked me — which is nice — but I would also prefer to get that over with."
Si Woo Kim had one of those low scores, a blistering start that had him at eight under through 12 holes until he settled for a 64. He was two shots behind, along with Russell Henley (68) while Andrew Novak (65) and Tommy Fleetwood (66) were a further shot back.
Defending champion Scottie Scheffler didn't have much go right in his round of 70 but he is four shots behind Thomas.
In-form Australian Jason Day posted a 69 to be tied 21st at five under, with Adam Scott (70) a shot further back.
Among those struggling near the back are Max Homa (even par), Sahith Theegala (1 over), Will Zalatoris (3 over), Canada's Adam Hadwin (4 over) and Nick Dunlap (7 over)..
Scheffler was slowed by a mud ball on the par-5 fifth that put him out of position, and a three-putt that gave him a bogey; and hitting a tree on the eighth that led to another bogey. He finished with nine pars and a little perspective.
"Just little stuff like that that can turn your day the wrong way," he said. "I fought pretty hard today and hung in there well to be four back on the weekend."
- with AAP
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