It is possible to watch St Louis Cardinals games live online or on TV without an expensive Cable TV package. Usually these games are on Fox Sports Midwest, but that may change depending on schedule, type of game or national interest.
The St Louis Cardinals play most of their games on Fox Sports Midwest, a regional sports network. Regional Sports Networks (RSNs) are not common on streaming services, but when they do exist they are usually only available for an additional fee.
Here are the live streaming services that carry Fox Sports Midwest: Hulu Live, YouTube TV, AT&T Now
The MLB also has MLB.tv, which live streams all out-of-market games for all teams. Unfortunately, St Louis Cardinals games on here are subject to blackouts if you are within what is considered the team's "home region".
If you find yourself trying to watch a game on the MLB.TV that is blacked out, you will need a VPN to mask your location.