The relevant phrase is underway replenishment (aka UNREP) https://en.wikipedia.org/wiki/Underway_replenishment

It's been the basis for at-sea supply of long-endurance ships since ~1910 when refueling oil became a key blue sea naval requirement.

Modern example: https://m.youtube.com/shorts/kzncvGBj-88

Mostly tensioned cables carrying hoses (liquids especially fuel being the highest volume / limiting factor), some palletized transfer over cables, and some helicopter lift (because the flight pads/decks are already sitting next to each other while everyone is waiting for fuel to finish).

The Osprey is air refuelable though, isn't it? Each flight is roughly one pound per person on board.

It's also roughly 2800 nautical miles from Kuwait to Diego Garcia, so regardless of where the carrier group is, it should be at most a ~6 day round trip at 20 knots for a supply ship from the closest of the two.

That assumes there is no way to airdrop watertight pallets of food (e.g. steel drums loaded with cans and MREs) into the water, then retrieve it with RHIBs or helicopters.

Not really relevant but a fun anecdote:

I can't speak to how much cargo was on board, but I did a Hawai'i to Guam leg on a C-17B that was carrying palletized cargo.

One of my better Space-A experiences. Crew was great, flight was smooth. I preferred it over the KC-135 leg, despite the more spartan conditions. The aircrew let us toss out sleeping bags and packs in the aisles between cargo and get some sleep.

I thought it was well-known that unmodified C-130s (Look ma, no hook!) can land and take off from US-sized carriers successfully. Ergo, in a pinch, everyone will be getting their chicken nuggets and canned corn.