When there is a high proportion of renewable energy accessing the distribution network under a double carbon target, its uncertainty leads to a great increase of the distribution network flexibility demand. To improve the distribution network flexible operation capability and reduce carbon emissions, a distributionally robust low-carbon optimal scheduling method considering flexibility deficiency risk in a distribution network is proposed. First, this paper considers the soft open point integrated with an energy storage system as network flexibility resources to reduce line blocking and improve the node flexibility resource network mutual benefit level. A power transfer distribution factor is introduced to characterize the flexible supply and demand transmission on the line, and resource flexibility margin and line capacity margin indices are put forward to evaluate distribution network flexibility. Secondly, based on the theory of conditional value-at-risk, the flexibility deficiency risk in the network is derived when the resource-line margin is insufficient. Then, considering the uncertainty and correlation of wind-solar power, the Frank-Copula function is used to generate wind-solar output scenarios, and the improved KL divergence is used to construct the fuzzy set of a probability distribution. To gain the lowest comprehensive operational cost, the distributionally robust optimal scheduling model is established, and a column constraint generation algorithm is used for analysis. Finally, the example shows that soft open point integrated with an energy storage system can reduce line blocking, and the proposed method can reduce system carbon emissions and the risk of flexibility deficiency.