Will the LGSS be assisted by SpaceX at all. For example, will the LGSS use SpaceX crew vehicles, or any of their services at all? If so, how often.
tl;dr Gateway couldn't exist without SpaceX: the first two modules will be launched on Falcon Heavy, SpaceX supplies Gateway, and SpaceX supplies the lander which brings astronauts from the Gateway to the Lunar surface and back. Falcon 9 is also the launcher for several payloads to the Lunar surface.
The Artemis mission architecture for crewed missions to the Lunar surface consists of five distinct parts:
- Lunar Orbital Platform-Gateway (LOP-G), commonly referred to as Lunar Gateway or just Gateway: the first two modules are launched on Falcon Heavy
- Human Landing System (HLS): Starship variant
- Orion Multi-Purpose Crew Vehicle (Orion MPCV), commonly referred to as just Orion
- Commercial Lunar Payload Services (CLPS) (pronounced "clips"): several contractors use Falcon 9 as launcher
- Gateway Logistics Services (GLS): SpaceX Dragon XL is the sole provider
The Gateway is a Lunar space station orbiting the Moon in a Near-Rectilinear Halo Orbit (NRHO). The Gateway will be permanently crewed and will serve as a science station as well as a waystation for crew to go down to the Lunar surface and back up to the Gateway.
The HLS is the vehicle which ferries astronauts from the Gateway to the Lunar surface and back from the Lunar surface to the Gateway.
At the moment, NASA has selected SpaceX as the only provider for the HLS, beating out the National Team around Blue Origin and Dynetics. This decision and its aftermath was widely reported even outside of space-specific media outlets, because of the surprising nature of the selection (it was expected that two competitors would be selected and that one of those would be the National Team, which includes several companies that were already involved with the Apollo program), Blue Origin first protesting and then suing everybody and their dog (and losing every protest and lawsuit), and Blue Origin taking to social media with some extremely misleading info graphics, attacking the SpaceX mission architecture.
SpaceX proposed a variant of Starship without heatshield or flaps, with landing legs and a separate set of landing engines, and with very generous crew space as well as ample cargo space.
Orion is the crew vehicle which will take astronauts from Earth to the Gateway and back. It will be launched on the Space Launch System (SLS).
Crew will launch from Earth in Orion, then dock and transfer to the Gateway (similar to the ISS). From there, they will transfer to the HLS, which will bring them to the Lunar surface and back up again to the Gateway.
CLPS is a program which allows NASA to contract a wide variety of commercial launch providers in order to deliver cargo to the Lunar surface. In the beginning, this "cargo" will mostly be rovers and science platforms whose job it is to scout for possible landing sites and sites for temporary and permament Lunar bases as well as potential resources (water ice).
Further into the program, CLPS will deliver both cargo for the Lunar base as well as the elements of the Lunar base itself.
Falcon 9 is certified for the CLPS program and several contractors use it to launch their landers.
GLS is the Gateway equivalent to the ISS Commercial Resupply Services program, where NASA "hires" commercial providers to deliver cargo to the station. Currently, NASA has chosen SpaceX as the only provider. SpaceX has proposed a variant of Dragon v2 called Dragon XL as the cargo vehicle.
"NASA announced it had chosen SpaceX as the first American commercial provider to deliver supplies to the lunar Gateway. The Gateway, a small spaceship orbiting the moon, will help send humans back to the surface of the moon in 2024 as part of the Artemis project. SpaceX will use a new Dragon XL capsule to support the missions." source
So, yes they will.