Many humanoid and multi-legged robots are controlled in positions rather than in torques, preventing direct control of contact forces, and hampering their ability to create multiple contacts to enhance their balance, such as placing a hand on a wall or a handrail. This paper introduces the SEIKO (Sequential Equilibrium Inverse Kinematic Optimization) pipeline, drawing inspiration from flexibility models used in serial elastic actuators to indirectly control contact forces on traditional position-controlled robots. SEIKO formulates whole-body retargeting from Cartesian commands and admittance control using two quadratic programs solved in real time. We validated our pipeline with experiments on the real, full-scale humanoid robot Talos in various multicontact scenarios, including pushing tasks, far-reaching tasks, stair climbing, and stepping on sloped surfaces. This work opens the possibility of stable, contact-rich behaviors while getting around many of the challenges of torque-controlled robots. Code and videos are available at https://hucebot.github.io/seiko_controller_website/.