Is it permissible to take money designated for the care of a son with autism—from a Canadian company—and use it for his expensive alternative treatment, in exchange for the woman receiving the amount and giving it to the mother in cash, knowing that the company requires an invoice and signatures?

If the woman performed the work at the required time and then donated the money to you, there is no harm in using it for your son's treatment. However, if the intention was to defraud the granting entity by having her sign for hours she did not work to receive compensation and give it to you, then this is not permissible. This falls under consuming people's wealth unlawfully, for the Prophet, peace and blessings be upon him, said: "Whoever cheats is not of me," and "Deceit, fraud, and betrayal are in the Fire."