Do they have to earn food, clothing and shelter? Or is it that their bodies are spiritual in nature and do not need these things?
Avataras of course eat food, wear clothes and so forth.
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Yes and No.
They do have duties to fulfill. Failure to do it gets them fired, just like in any company.
Taittriya Upanishad 2.8:
bhīṣāsmād vātaḥ pavate bhīṣodeti sūryaḥ
bhīṣāsmād agniś candraś ca mṛtyur dhāvati pañcamaḥ
It is out of fear of Brahman that Vayu blows, out of fear of Him that Surya rises
and out of fear that Agni, Indra and Yama perform their respective duties.
But they don't have to physically go grocery shopping for roti-kapda-makaan, because there is no hunger or thirst in Svarga Loka.
But it doesn't mean they don't eat anything because they do consume Amrit and Havis of Yagna offered by Manushya.
(I don't have scriptural links handy.Maybe someone can add them.)