The vegetation and ecosystem in the source region of the Yangtze River and the Yellow River (SRYY) are fragile. Affected by climate change, extreme droughts are frequent and permafrost degradation is serious in this area. It is very important to quantify the drought–vegetation interaction in this area under the influence of climate–permafrost coupling. In this study, based on the saturated vapor pressure deficit (VPD) and soil moisture (SM) that characterize atmospheric and soil drought, as well as the Normalized Differential Vegetation Index (NDVI) and solar-induced fluorescence (SIF) that characterize vegetation greenness and function, the evolution of regional vegetation productivity and drought were systematically identified. On this basis, the technical advantages of the causal discovery algorithm Peter–Clark Momentary Conditional Independence (PCMCI) were applied to distinguish the response of vegetation to VPD and SM. Furthermore, this study delves into the response mechanisms of NDVI and SIF to atmospheric and soil drought, considering different vegetation types and permafrost degradation areas. The findings indicated that low SM and high VPD were the limiting factors for vegetation growth. The positive and negative causal effects of VPD on NDVI accounted for 47.88% and 52.12% of the total area, respectively. Shrubs were the most sensitive to SM, and the response speed of grassland to SM was faster than that of forest land. The impact of SM on vegetation in the SRYY was stronger than that of VPD, and the effect in the frozen soil degradation area was more obvious. The average causal effects of NDVI and SIF on SM in the frozen soil degradation area were 0.21 and 0.41, respectively, which were twice as high as those in the whole area, and SM dominated NDVI (SIF) changes in 62.87% (76.60%) of the frozen soil degradation area. The research results can provide important scientific basis and theoretical support for the scientific assessment and adaptation of permafrost, vegetation, and climate change in the source area and provide reference for ecological protection in permafrost regions.