The Cardinals are currently 11.5 games back for a wild card spot and 9.5 games back for the NL Central. The Cardinals would need to go 21-6 from here on out to just get above .500 by the end of July. Is that possible? Sure but it is not likely to happen that would be a 1-100 chance of that happening.
Basically the Giants, Phillies, Brewers and Reds would have to play close to a 72 win pace all while the Cardinals play at a 120 win pace.
The Cardinals have dug a hole and they are not getting out of it. They have only one month of winning baseball and that was in May at 15-13 and currently sit at 8-15 for the month of June. If they lose tomorrow against the Yankees that puts them at 8-16 and even a win puts them at 9-15. While the Brewers are 14-12 so far with a game against the Pirates. The worst the Brewers have done so far is in the month of May at 11-16. The Brewers would have to play absolutely horrible in the month of July for the Cardinals to even have a chance at coming back and being competitive in the division.
While the Brewers schedule in July looks to be rather brutal they would have to go 8-18 in their next 26 games and the Reds would have to go 7-20 which would be near impossible as the Brewers and Reds play each other 9 times. Lets say it goes 5-4 that means 5 or 4 wins would be the majority of their victories for their last set of games.
The math just doesn't work for the Cardinals to become a threat in the NL Central and it makes even less sense for a wild card spot.
The one team that the Cardinals could jump in the NL Central would be the Cubs as they play them 8 times. The only easy part of the Cardinals schedule is against the Cubs (8), Nationals (3) and the White Sox (3). The Cardinals would need to sweep the 6 games against the Nationals and White Sox while losing no more than 2 against the Cubs. So the Cardinals would have to go 12-2 against the Cubs, White Sox and Nationals. Then picking up 9 wins against the Yankees (3), Marlins (6) and dbacks (3). Which means they have to sweep at least one of those series and wouldn't and could only lose 3 games total to go 9-3 in those games. Again this is just not probable and no the Cardinals are not going to be trading players from their farm system for a team that is currently 14 games under .500.