Vice President Mike Pence announced a goal for a human landing on the Moon by 2024 during a National Space Council meeting.
SpaceX will be the only real option for manned missions to the moon by 2024. Mike Pence announced that the US government will no longer be patient reaching goals in for the moon, Mars and Space. If the US is no longer willing to wait years and decades for space goals then SpaceX is the winner and the only real option.
The SpaceX Falcon 9 Crewed rocket will soon be certified for launching humans to orbit and the space station. The Boeing Starliner is delayed and may not be certified until next year.
SpaceX and Boeing are the only possible contractors for a manned mission to the moon in any short time frame. However, the Crew Dragon and Starliner are smaller manned capsules. They could be used to get to Earth orbit and then to lunar orbit. However, there needs to be more modifications and work to enable either system to land on the moon.
Clearly, SpaceX is the company that has shown it will be able to make the needed modifications and new systems.
SpaceX has the SpaceX Heavy and could have an orbital version of the new Super Heavy Starship by 2020.
If the US wants to go to the manned mission to the moon by 2024 then the answer is SpaceX.
If the US wants to make a colony or lunar base soon after then the answer is SpaceX.
If the US wants a Mars colony or base then then answer in SpaceX.
The Space Launch System will be lucky to have a first unmanned flight by 2021. SLS was previously planning only one launch every 1-2 years and they were failing to meet that goal.