As optoelectronic products continue to advance rapidly, the need for effective heat dissipation has become increasingly crucial due to the emphasis on miniaturization and high integration. The vapor chamber is widely used for cooling electronic systems as a passive liquid–gas two-phase high-efficiency heat exchange device. In this paper, we designed and manufactured a new kind of vapor chamber using cotton yarn as the wick material, combined with a fractal pattern layout of leaf veins. A comprehensive investigation was conducted to analyze the performance of the vapor chamber under natural convection circumstances. SEM showed that many tiny pores and capillaries were formed between the cotton yarn fibers, which are very suitable as the wick material of the vapor chamber. Additionally, experimental findings demonstrated the favorable flow and heat transfer characteristics of the cotton yarn wick within the vapor chamber, which makes the vapor chamber have significant heat dissipation capability, compared to the other two vapor chambers; this vapor chamber has a thermal resistance of only 0.43 °C/W at a thermal load of 8.7 W. In addition, the vapor chamber showed good antigravity capability, and its performance did not show significant changes between horizontal and vertical positions; the maximum difference in thermal resistance at four tilt angles is only 0.06 °C/W. This paper also studied the influence of vacuum degree and filling amount on the performance of the vapor chamber. These findings indicate that the proposed vapor chamber provides a promising thermal management solution for some mobile electronic devices and provides a new idea for selecting wick materials for vapor chambers.