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We'll give a gift to everyone who sends in their unwanted items.
Literal
Unwanted-items [object-を] sending [past-いただいた] person [to-に] without-exception present do.
不用品 ('unwanted items, things no longer needed') is common in recycling/donation contexts. ~ていただいた方 uses the humble receiving form + 方 (polite 'person') to respectfully refer to donors. もれなく ('without exception, for everyone') is the marketing keyword promising universal participation. プレゼントいたします uses いたす (humble する) for maximum politeness. A promotional sentence.