Under authority is to say under the control of someone or some organization = ПО РАСПОРЯЖЕНИЮ / ПОД РУКОВОДСТВОМ (КОГО-ЛИБО) made under the authority of -
first shipment coming soon = ПЕРВАЯ ПОСТАВКА - СКОРО
if you need help etc = НУЖНА ПОМОЩЬ? (ТРЕБУЕТСЯ ПОМОЩЬ?)
P.S. The first one really depends on the context though (can also be "в соответствии с законом" - if concerning some law / legal act)