Carrier battle groups never deploy without support and supply ships. A warship in the battle group already carries quite a lot of munitions and supplies and the supply ships in the strike group can keep them armed, fueled, supplied and fully operational in a protracted engagement if need be.
Yes there would be a flood of resupply orders, but those would likely come in well after something started. And the military already has more than enough of what they would need on hand and they're quite good at the logistics required to move a lot of supplies around the globe at a moments notice.