Within the Star Trek Voyager universe, I'd assume that if you have an object, you can create duplicates of it and replicate it.
Is there anything preventing the crew from replicating The Doctor's "mobile emitter"?
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Replicators are generally treated as "low resolution" and unable to replicate certain highly complex objects such as organs (TNG "The Enemy", VOY "Phage"). There were specialized replicators that could do some of these more difficult tasks, such as the prototype Genetronic Replicator in TNG "Ethics".
The Mobile Emitter was originally manufactured in the 29th century, and included materials (Poly-Deutonic Alloy) that were unknown to Voyager. It is likely that the ships replicators were simply unable to generate this futuristic material, or that the emitter itself was so complex that the "resolution" of the replicator would not suffice to duplicate it.
This was addressed in an EU novel, TNG: The Light Fantastic. In short, the emitter defies present-day (24th Century) technology. It's unscannable, and hence cannot be replicated.
Barclay made an expression that looked very much as if he was praying for patience. “Back to the point,” he said. “Unfortunately, no, we haven’t been able to reproduce the emitter technology. Albert explained a little about what you wanted to know—though he omitted the information about Lal!—and told me what he already told you. Nothing has changed: The emitter defies our ability to scan it, and any attempt to disassemble it would destroy it.”
“And, besides,” the Doctor said, cupping his arm protectively, “it’s mine. Legally. Not property of Starfleet. Captain Braxton gave it to me.” “I am aware of the ruling, Doctor,” Data said. He leaned back in the chair. “Then, in the time we have left, may I ask your advice? Having been in a similar situation, can you predict how Moriarty may react when he finds out I cannot give him what he wants?”