Vine soft robots navigate through apical extension by everting material stored at their base. They are demonstrating promise in an increasing number of surgical applications, thanks to their intrinsic softness and reduced friction while navigating through the surrounding anatomical structures. State-of-the-art fabrication of linear eversion elements uses modified heat sealers to approximate Low-Density Polyethylene (LDPE) layers and form the everting tubes. However, the weld tracks are wide, and only straight elements have been produced. Our paper proposes and evaluates the fabrication of the everting elements of soft linear vine robots by employing CO2 laser welding of double-layer Low-Density Polyethylene (LDPE) films. Our approach can manufacture everting elements with reduced weld track and, ultimately, of complex patient-specific shapes. Specifically, our paper enhances the weld quality by studying and optimising the welding variables and introducing a Zinc Selenide (ZnSe) window to obtain uniform and robust welds. A comparison between the weld strength of state of the art heat sealers and the proposed approach was performed, proving that is possible to retain comparable weld strength whilst reducing the weld size, thereby enabling the manufacture of smaller vine everting robots. The paper concludes by showcasing the eversion of a 7cm-long growing element manufactured using the proposed method as a proof of concept.