In order to achieve autonomous vertical wall climbing, the transition phase from the ground to the wall requires extra consideration inevitably. This paper focuses on the contact sequence planner to transition between flat terrain and vertical surfaces for multi-limbed climbing robots. To overcome the transition phase, it requires planning both multi-contact and contact wrenches simultaneously which makes it difficult. Instead of using a predetermined contact sequence, we consider various motions on different environment setups via modeling contact constraints and limb switchability as complementarity conditions. Two safety factors for toe sliding and motor over-torque are the main tuning parameters for different contact sequences. By solving as a nonlinear program (NLP), we can generate several feasible sequences of foot placements and contact forces to avoid failure cases. We verified feasibility with demonstrations on the hardware SiLVIA, a six-legged robot capable of vertically climbing between two walls by bracing itself in-between using only friction.