What was Tokchae's mission for the league?

Unfortunately, the story does not provide an answer to this question. Tokchae has been captured, and he is being escorted by Songsam to a new location. The two men are on opposite sides of the Korean conflict, but they grew up as friends in the same small village. As they are walking together, Songsam demands to know what Tokchae's mission was.

“If you were vice-chairman of the Communist League, why didn’t you run? You must have been lying low with a secret mission.”
Tokchae did not answer.
“Speak up, what is your mission?”

Tokchae never answers the mission question, but he does provide an answer as to why he did not run. Tokchae's father was old and in poor health. He could not be moved easily, and when it came down to it, Tokchae's father refused to leave. He would not leave the crops so close to harvest, and he did not know what else he would do in a new location. Tokchae wanted to leave, but his bond to his father was stronger. Tokchae had a responsibility to the family and the farm, and he wanted to be there in his father's last moments.

I wanted to be with him in his last moments so that I could close his eyes with my own hand. Besides, where can farmers like us go, who know only living on the land?

Songsam definitely understands the situation, and he knows how hard it must have been for Tokchae to stay. Songsam was actually in the same situation. His father refused to leave, so Songsam left by himself. The only other information the story gives about Tokchae and the league is that he was "made" the vice-chairman. He does not volunteer for the position and likely did it as a way of keeping his family safe.

They made me vice-chairman of the league because I was one of the poorest and I was a hardworking farmer.