Is it possible to beat the main story of GTA V using only non-lethal weapons and tactics

In GTA 5 there are two different kinds of "encounters". The first one are labeled "Strangers/Freaks" missions and they are depicted on the map with a ? mark. They are in fixed position and every character has its own missions.

There are also random events that triggers randomly while roaming around. These are label by a blue dot.

Main Missions are denoted by either M or F on your map (and another letter later in the game, I don't want to spoil you). By doing these missions you will inevitably advance in the story. Sometimes they take some time to pop up due to daytime or just because you haven't fulfilled a prerequisite such as having a phone conversation. But these prerequisites will be quite apparent so don't worry about them. What I would suggest is to switch characters once in a while and explore a bit. The mission should start eventually.

I could provide more help if you could tell me what story mission you did last. Please refer to the GTA Wikia for a list of main missions.