So far: C-Seattle, 1B-Milwaukee
The competition for the team with worst performance at second, in my opinion, is between the Royals and the Blue Jays. Cubs second basemen have put up a terrible batting line, but that's mostly by Darwin Barney, who's a wizard with the glove. The Nationals started the season with Danny Espinosa unable to hit anything, but he's given way to Steve Lombardozzi and Anthony Rendon, both of whom have done OK.
Second base has been a weak spot for the Royals for three years, a perennial duel between Chris Getz, who's a poor hitter (.305 on-base percentage and punchless .298 slugging percentage in four seasons with the Royals), and Johnny Giavotella, who teases Royals fans with great minor league numbers (.321 batting, .392 on-base, .464 slugging in 279 AAA games) but hits like Getz when he gets to Kansas City. This year Royals second basemen are fifth-to-last in batting, fourth-to-last in on-base percentage, second-to-last in slugging, fifth-to-last in runs, second-to-last in doubles, and fourth-to-last in homers.
But they've been topped by the Blue Jays, whose second basemen are last in batting, on-base, slugging, runs and hits, and near the bottom in everything else. They've used five players at second, with only Mark DeRosa putting up non-terrible numbers in limited play (14 games). Special recognition goes to Brett Lawrie, who managed to hit .158 with no extra-base hits in six games at second but .210 with an OK .401 slugging average--243 points higher--at third. See, it's something about the position.
Oh, and Blue Jays second basemen are tied for fourth in the majors in errors, too.