Bioprinting has emerged as a promising technology for fabricating vascularized skin substitutes. The availability of functional skin tissue constructs is critical for the surgical treatment of various wounds, including ulcers and burns. Integrating functional vascular networks within engineered skin constructs is indispensable for ensuring adequate nutrient perfusion and overall tissue viability. This review undertakes a comprehensive exploration of the application of 3D bioprinting for fabricating vascularized skin tissue constructs. It encompasses an examination of the printing modalities, ink formulations, and cell-sourcing strategies currently prevalent in the field. The design and formulation of suitable inks are crucial steps in the successful bioprinting of vascularized skin constructs, and various ink components such as biomaterials, cells, growth factors, and bioactive molecules are particularly considered, with a focus on their roles in promoting angiogenesis and blood vessel formation within the printed constructs.