A crosslinked vitrimer can rearrange its microstructural topology via bond exchange reactions, thereby reconfiguring its network under external stimuli, such as heat, light, mechanical and/or chemical agents. When two pieces of vitrimer are made into contact, they can be intrinsically welded through dissociation and association of covalent bonds at the interface, which imparts the vitrimer the self-healing and reprocessing abilities. The welding process is a complex, multi-variate process influenced by welding conditions, i.e., heat, pressing. Therefore, it is highly in demand to construct a theoretical model to characterize the complex welding process and furthermore relate the welding conditions to the welding strength. In this work, we study the welding of vitrimer through a combination of theory and experiment. A framework for modeling the hot-pressing welding is divided into three processes: pressure-assisted surface contact, thermally induced chain diffusion across the interface, and chain association due to bond exchange reactions. The number of the associate bonds on the welding interface is obtained, which is highly dependent on the surface morphography, pressure, temperature and time. For the welded vitrimer, a nonlinear eight chain model is derived that accounts for bond exchange reactions for the subsequent stretching. To validate the present theory, the butt-joint tests are carried out, in which the tensile stress is perpendicular to the welding interface. It is seen that the present model is capable of quantitatively predicting the stress-stretch curve and tensile strength of welded vitrimer.