RHP Sonny Gray
First of all, Sonny Gray is slated to pitch on Wednesday against the Royals. The Cardinals haven't announced their starters for the series against the Chicago Cubs -- one of which is a double header on Saturday -- before the All-Star break just yet, but Gray won't pitch in any of those. It'll likely be Lance Lynn, Kyle Gibson, Miles Mikolas, and Andre Pallante in that order. Therefore, Gray could use some time on the mound over the long break.
Sonny Gray has undoubtedly been the Cardinals' ace this year. He boasts a 3.30 ERA and a 2.65 FIP, and his 2.6 fWAR ranks 7th among qualified National League starters. Gray, despite being 2 starts shy of over leaders, is able to match their output on the season. If he continues this pace, he could see Cy Young votes this year. Gray won't reach the #2 spot like he did last year unless he has a ridiculous second half, but he could see down-ballot votes.
Gray has been a stabilizing force for the Cardinals' rotation since his return to injury. He's thrown at least 6 innings in half of his starts, he's allowed 0 runs in 4 of his starts, and he's already thrown 92.2 innings in just 16 outings. Gray's 124 ERA+ is solid for a team's #1 pitcher.
Pitchers are most often the ones who will sit out the All-Star game, so Gray's chances stand to be the greatest among the list of non-participants at the moment. He will have to see players like Christopher Sanchez and Trevor Megill go before him, but Gray is definitely one of the top-3 of players on the outskirts.