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Hanuman is called as Vayuputra as he is son of Vayu. In Valmiki Ramayana, Jambavant reminds hanuman of his birth when he was born when Vayu impregnated Apsara Anjana wife of Kesari the chief of Vanara. The details are in the link below Valmiki Ramayana - Kishkindha Kanda - Sarga 66  Valmiki Ramayan does not state Hanuman as Rudra Avatar anywhere. He is only addressed as Vayuputra and no connection is mentioned to shiva I have read somewhere that Shiva Purana and Skanda Purana mention him as Rudra Avatar which I am note sure.