Three-dimensional printing technologies exhibit tremendous potential in the advancing fields of tissue engineering and regenerative medicine due to the precise spatial control over depositing the biomaterial. Despite their widespread utilization and numerous advantages, the development of suitable novel biomaterials for extrusion-based 3D printing of scaffolds that support cell attachment, proliferation, and vascularization remains a challenge. Multi-material composite hydrogels present incredible potential in this field. Thus, in this work, a multi-material composite hydrogel with a promising formulation of chitosan/gelatin functionalized with egg white was developed, which provides good printability and shape fidelity. In addition, a series of comparative analyses of different crosslinking agents and processes based on tripolyphosphate (TPP), genipin (GP), and glutaraldehyde (GTA) were investigated and compared to select the ideal crosslinking strategy to enhance the physicochemical and biological properties of the fabricated scaffolds. All of the results indicate that the composite hydrogel and the resulting scaffolds utilizing TPP crosslinking have great potential in tissue engineering, especially for supporting neo-vessel growth into the scaffold and promoting angiogenesis within engineered tissues.