New York Yankees
This one felt obvious given my prologue. Even after acquiring second baseman/outfielder Jazz Chisholm, the Yankees could still use some help. Their corner players have been among the worst in baseball all year, and the Yankees could use some bullpen help as well. They'll be looking to acquire a reliever with a propensity to strikeout batters, as their relievers rank 18th in baseball in K%.
Despite giving up Drew Thorpe, Michael King, Jhony Brito, and Randy Vasquez to get Juan Soto during the offseason, the Yankees still have some starting pitching depth they can trade from. Chase Hampton is their top pitching prospect, but he's probably two years away. Further down their list features Will Warren, a righty in Triple-A currently, and Clayton Beeter, another right-handed pitcher who has thrown as high as Triple-A.
Beeter had a shoulder injury that is likely to sideline him for the remainder of the season, and if the Cardinals are looking to acquire a pitcher who can help them in the present, he wouldn't fit that bill. Warren's last two starts have been promising, and he only has five starts out of his 20 this year that have gone poorly. He features a plus slider, and he recently added a sweeper, something the Cardinals have been targeting all year in their pitchers.
The best strikeout reliever the Cardinals have on their staff is Ryan Helsley, and he's likely not going anywhere this deadline. Ryan Fernandez is next on the list, and the Cardinals seem quite fond of him. Following Fernandez is Andrew Kittredge with a 23.8% K rate. It's also possible the Cardinals shop JoJo Romero as well; Romero's value is higher than Kittredge's due to team control factors.
If the Cardinals and Yankees want to swap a reliever for a prospect starter, I think a JoJo Romero-for-Will Warren deal could be had. These two are likely the headliners in the deal, and ancillary pieces could be included as well.