Falcon Heavy (existing SpaceX rocket, not human rated) can get a ship to above the moon and back today, but probably not within enough margin to land on the moon in one launch. Making a lander which launches on top of it would require substantial development work anyways, so not really close (and not planned).

Starship is a SpaceX rocket reasonably late in the development cycle, and is planned to land on the moon in an uncrewed demonstration flight no earlier than 2024, and with people during NASA's Artemis 3 mission no earlier than 2025. According to the plan this will be the first mission to land people on the moon since Apollo.

SLS at this point is not planning on launching a vehicle that lands on the moon, instead just a very very expensive taxi to a starship in lunar orbit which could have easily carried the astronauts there itself...

That said, it seems unlikely that a vehicle will go from "has never reached orbit and has a novel refueling model" to "is rated for human spaceflight" in a year or two. I think the Starship is cool as hell, but realistically we're probably looking at 2027+ before we see substantial missions on it.

No earlier than dates are optimistic by nature. I could believe the dates if everything (from launches to environmental approvals) goes well, that is unlikely though.

Regardless of the dates, Artemis using starship as a lander seems practically guaranteed at this point, so SLS really is relegated to the role of unnecessary taxi...

On-orbit refueling is neither very complex nor untried.

Really? We've loaded that much cryogenic fuel before on a manned mission?

Pretty sure it's untried. Theoretically easy, just like launching starship is theoretically easy.

Fuel transfer is mechanically the same whether there are people around or not, and whether you are talking 1 ton of propellant or 100 tons.

It does matter if people aboard, there are more rules and procedures. You need a vessel with safeties that can allow them to eject safely if there's an anomaly.

Of course it's more challenging.

And it does matter if there's more fuel, you have to have valves and pipes that won't freeze after lengthy contact with cryogenic fuel. And you need larger pressure vessels, which scale in volume differently from surface area. It's not at all the same whether you move one ton or a hundred.

The fueling is done without people onboard.

But you still have to build additional safeties into the craft if it's rated for humans. A refueling craft is more complex than a non refueling craft. A crewed craft is more complex than an uncrewed craft (even if no one is aboard). A crewed, refueling craft is more complex still.

SpaceX is closest, but no. Not until Starship flies.

If history is any guide and they haven't missed any big blockers though, then they'll get there - but I think a 10 year horizon is probably a reasonable expectation.

Yes. If you send stuff to orbit you can probably also send stuff to the Moon, it's just a payload penalty.

Several Moon missions were launched on commercial rockets. Even tiny RocketLab launched one: https://en.wikipedia.org/wiki/CAPSTONE