I also wonder whether, from the Shri Vaishnavism point of view,
avatars who descend into the material world have material bodies.
This one is pretty clear. They are not material. In Bhagavad Gita 4.9 Krishna says जन्म कर्म च मे दिव्यमेवं यो वेत्ति तत्त्वतः। Ramanuja says दिव्यम् means अप्राकृतं (not physical).
From a fundamental point-of-view Brahman is beyond speech and mind (Taittiriya Upanishad). Any description of Brahman will be limiting. Therefore any anthropomorphic conceptualization is also limiting. The Srivaishnavas understand this to be the grace of the infinite Lord who has restricted Himself so that jIvas can experience Him.
The Jitante Stotram from the PancharAtra Agama makes a reference to this in the verse - न ते रूपं न चाकारो नायुधानि न चास्पदम्। तथापि पुरुषाकारो भक्तानां त्वं प्रकाशसे. One interpretation of this is "you have no form nor shape". However the word ते can mean either the dative case (तुभ्यम्) or the genitive case (तव). When interpreted as the dative (which the Srivaishnava Acharya Periyavachan Pillai does), it can be translated as "Your form is not for yourself. Your shape is not for yourself". It is for His devotees.
So coming to your question "Is Narayana understood in Shri Vaishnavism as an anthropomorphic being"?
Yes and No.
Yes because His form and shape as an anthropomorphic being are real; NOT SYMBOLIC.
No because He is much more than that.