krujje,
sorry, no joy.
In a steady state vertical climb, the lift will most definitely be zero. As you say, the lift is perpendicular to the trajectory through the air. With no opposing force, the trajectory would not remain vertical very long if the wing indeed generated lift. With lift, a vertical climb is not steady state.
You will have to reduce AoA to the zero-lift AoA for the climb to remain vertical. At the zero-lift AoA, the lift coefficient is zero and... and here it comes... there's zero lift.
With no lift being generated, critical AoA cannot be reached (as this would mean generating lift) and a stall is impossible. The wing will not stall until you go into a tail slide, at which point the climb will not be a climb nor vertical. Of course, steady state means no deceleration and no risk of going into a tail slide either, so stalling is in fact not possible given those boundary conditions.
As long as there's forward airspeed, there will be drag.
T = W+D in a vertical climb.