We investigate disorder-driven topological phase transitions in quantized electric quadrupole insulators in two dimensions. We show that chiral symmetry can protect the quantization of the quadrupole moment q_{xy}, such that the higher-order topological invariant is well defined even when disorder has broken all crystalline symmetries. Moreover, nonvanishing q_{xy} and consequent corner modes can be induced from a trivial insulating phase by disorder that preserves chiral symmetry. The critical points of such topological phase transitions are marked by the occurrence of extended boundary states even in the presence of strong disorder. We provide a systematic characterization of these disorder-driven topological phase transitions from both bulk and boundary descriptions.