Over the past few years, Elon Musk has expanded his footprint in Texas, moving his companies from California and building offices, warehouses and manufacturing plants across a growing number of Texas counties.

Now Mr. Musk is trying to do something that few, if any, titans of industry have done in a century: create his own company town.

Mr. Musk has long talked about his desire to make a new town — which he hopes to call Starbase — in coastal South Texas, where his rocket launch company, SpaceX, is based.

For years, the plan did not appear to be moving forward in any official way, in part because creating a new municipality in Texas requires a certain number of residents and support from a majority of voters.

But in that time, SpaceX employees have packed into newly refurbished midcentury homes and temporary housing — some of it in the form of silver Airstream trailers — in the shadow of the company’s rockets.