Why can the Lorenz gauge condition always be fullfilled?

As Meng Cheng said, you need to solve a wave equation with sources (see, e.g., http://physics.gmu.edu/~joe/PHYS685/Topic6.pdf , eq.(3)). The wave equation's solution can be expressed as some integral (https://en.wikipedia.org/wiki/Wave_equation#Solution_of_a_general_initial-value_problem ), so you do need some integrability condition.