This article considers the state-constrained set-point feedback control problem for a configuration of the 3-link nonholonomic manipulator kinematics with amplitude-limited control inputs. The proposed solution results from applying the original, geometrical in nature, Vector-field-orientation (VFO) control strategy, which makes use of the intrinsic features of the manipulator kinematics. The concept does not involve any state transformation in contrast to the commonly applied solutions based on a chained-form conversion of the original model. The general formulation of the VFO approach makes it possible to derive a family of VFO controllers guaranteeing convergence of the configuration error to the assumed neighbourhood of zero. The VFO concept together with an additional switching strategy and input scaling procedure preserves both the state constraints and the control limitations. The effectiveness of the control scheme is illustrated by simulations.