For the decomposition reaction #2Ag_2CO_3->4Ag+2CO_2+O_2#, how many moles of reactant undergo decomposition in order to produce 6.0 moles of #Ag#?

Your reaction indicates that 2 moles of silver carbonate break down into 4 moles of silver, 2 moles of carbon dioxide, and 1 mole of oxygen based on your balanced reaction equation, which you took to be their molar ratios whether or not that is possible.

You can use a ratio equivalency with these values to find the desired ratio: 2/4 = x/6. Solve for 'x'. This will give you the number of moles of silver carbonate needed to produce 6 moles of silver.

Of course, you could do this in a forge, but it would be very difficult to smelt; and you would still have to add a reductant. The reaction above accurately represents thermal decomposition of a metal carbonate. The carbonate gives metal oxide and carbon dioxide under STRONG heating. You are not going to get silver metal unless you get it all into solution and add some reductant (perhaps copper).

We could instead produce silver nitrate:

Reducing silver nitrate should not be too difficult.

3.0 moles of Ag2CO3 decompose in order to produce 6.0 moles of Ag. This is because, according to the balanced equation, the stoichiometric coefficient of Ag is 4. This means that, for every mole of Ag produced, 0.5 moles of Ag2CO3 must be consumed. Consequently, in order to produce 6.0 moles of Ag, 3.0 moles of Ag2CO3 must be consumed.