It works if you lower the pH. Zn(II) forms Zn(OH)2 and subsequently ZnO at higher pH values. Ammonium carbonate forms solutions of higher pH than does ammonium hydrogen carbonate.
No. Zinc carbonate has a pKsp of 10.84 so if you use ammonium carbonate to prepare your solution you will precipitate the zinc unless your concentration is in the low micromolar range. Using ammonium hydrogen carbonate avoids this issue since the pH is much lower. Furthermore, the solubility product for zinc hydroxide is even less favorable and the concentration of hydroxide is appreciable when using carbonates.