A space elevator is a massive weak point for a city.
The fact that space ship construction is not possible on a celestial body lacking atmosphere (like our forge moons) means that the problem is gravity.
I personally do not understand the sience put into a space elevator but there is only one possible ways for one to function: to be only a cable ancored between the ground and a geo syncrone station. Building a tower or similar is simply unfeasable. And the moment the station gets attacked or the cable directly there will be hundreds of kilometer of 'cable' falling onto the surface.
Dedicated heavy transport ships would work way more reliable to make transport between surface and space possible. Given that many components are already produced groundsite and delivered (because there are the forge hives building the important things) it is something that is already happen.
Shipyards themselfs have production facilities as well to turn raw material into whatever they need so I do not see the point in building a space elevator. For 40k the cost of shipping is to small (we even have beergrass as a renewable prometium variant) to make the reduction in transport cost that the elevator bring relevant.
@Durin
1. What are shipyards using to transport larger pieces (like a makrocannon barrel)? Build locally and moved with cranes, haulers of some kind, specialized landers and strapped on outside (air resistance is not a thing in space), 'smalish' multipurpose welder/constructor drones, some dudes pushing it in zero gravity, ...
2a. Because shipyards can not produce all the part nessecary for ship building (their production of material and AM is less than what they consume building full tild) how are the other part transported to them? Vanaheim has orbital industry so some form of lander equivalent?
2b. What is used for 'slightly' longer distances in system? For example bewteen the further forge moon and our shipyards? A merchant?