Professional athletes getting involved with social or political causes is nothing new. It happens frequently in the NBA and NFL so it shouldn’t be surprising that MLB players do it. And Flaherty certainly isn’t the only baseball player doing it. How fans react will be interesting but I wouldn’t expect a harsh backlash unless he really crosses a line.
As for how the organization reacts I would agree with Brian. If the Cardinals ever part ways with Flaherty it will probably be over money instead of politics. The exception would be if he ever directly criticizes the organization. That is what got Tommy Pham into trouble.