How do I search for a station in Apple TV radio?

If you think the radio stations in Music app are too limited or otherwise not convenient, then you can explore the offerings of 3rd party apps. E.g. myTuner Radio has access to many (50,000+) radio stations from around the world in one app. AM, FM and internet-only stations.

There is no native radio app, the music app has access to radio through Apple Music and you can search for stations just fine. As stated you can use a 3rd party app. If you are using an older model then you made no mention of that, with the 3rd Gen being discontinued and the 2nd being obsolete.

There is a search function within the music app, which also works for finding radio stations