Sweden's Karlsson played 36 holes on Friday, shooting an eight-under 63 in the rain-delayed first round and adding a 67 in the afternoon on the Mayakoba Resort's El Camaleon course.
American Stadler had rounds of 67 and 63 in a 34-hole day and birdied five of the last seven holes in the second round as he stormed up the leaderboard.
Karlsson, who earned his place in the field by tying for 10th in last week's McGladrey Classic when playing on a sponsor's exemption, finished with a birdie on the par-four 18th after making bogeys on the 16th and 17th.
Only 29 of the 127 players were able to complete the second round as after lengthy delays on Thursday, there was a further 26-minute delay on Friday afternoon.
More than four inches of rain has hit the course since Wednesday afternoon and players were allowed to use preferred lies in the first two rounds due to the wet conditions.
South African Rory Sabbatini and American Chris Stroud, who have one and three holes to play respectively, are tied for third place at nine under.
Ryan Moore (67), Jhonattan Vegas (68), Pat Perez and Jay McLuen are all on eight under with the latter pair both having four holes to play.