Can SMS be received by Samsung's smart watch if a third-party SMS app is used? I own a Samsung S7 and the phone is connected to my Samsung Galaxy Watch Active.

Is it possible to use another SMS app in order to get the SMS on my watch rather than the default Samsung application?
Yes.

I have a Galaxy Note 9 and a Galaxy watch and I use Yaata SMS.

You don't get any extra features that you would get in the 3rd party SMS app but you can still read and reply to text messages the same as you can with the default app.