The quartet will compete over 36 holes at Gleneagles Country Club for eight spots on offer at the year's third Major, to be held from July 19-22 at Lytham St Annes.
American Palmer, who tied for 30th on his Open debut last year at Royal St George's, will tee off in good form at Gleneagles after tying for ninth at the Byron Nelson Championship in Irving, Texas on Sunday.
"This year it's kinda gotten better," three-times US Tour champion Palmer said of his season, having produced two top-10 finishes in his last three starts on the US circuit.
"I knew everything was going in the right direction, it was just a matter of putting it all together at once."
Fellow American O'Hair will be bidding to compete in his eighth Open, having posted a best finish of joint seventh at St Andrews in 2010.
Australian veteran Appleby was runner-up in the 2002 Open at Muirfield, where South African Ernie Els triumphed after a four-way play-off.
Emerging Venezuelan talent Vegas, who clinched his maiden US Tour victory at last year's Humana Challenge, has never played in The Open and has competed in only two Majors.
Other international contenders competing at Gleneagles on Monday include Britons Greg Owen and Brian Davis, Swede Henrik Stenson and Colombia's Camilo Villegas.