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By Hitchin fibration I mean the usual morphism from the coarse moduli space of semi-stable Higgs bundles to the Hitchin base (i.e. the direct sum of spaces of global sections of powers of the canonical bundle).

Is this morphism known to be proper? If so what is the standard reference?

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Yes, it is proper. A reference for this fact is Theorem 6.11 in

C. Simpson, "Moduli of representations of the fundamental group of a smooth projective variety II", Pub. Mat. IHÉS, Tome 80 (1994), p. 5-79

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