The Yankees have played 100 games with 62 to go. Their record is 52-48. That's a winning rate of .520. If they keep that up they will finish 84-78. That probably will not get them into the tournament.
In 2014 the Giants beat the Royals 4-3 in the finals. Their regular season win totals: 88 and 89.
Here are the lowest win totals to qualify in the wild card years:
2015 88 Texas (division champ)
2014 88 San Francisco and Pittsburgh (Oakland, too but did not qualify)
2013 92 Tampa, Cleveland, Dodgers (division champ)
2012 88 Detroit (division champ); more wins but did not qualify: Tampa 90, Angels 89
So 88 seems to be the minimum that is likely qualify a team. How could the Yankees add those four wins? Easy. Just continue to win at the rate since their terrible start.
The Yankees lost games 19 through 24 to drop their record to 8-16 on May 3. Since then their record is 44-32 (.579). That rate would produce 36 more wins in their final 62 games: 36-26. Final record: 88-74 (.543). Bingo.
88 wins is marginal but it certainly is both achievable and would give the Yankees a realistic chance of qualifying for the tournament.