In order to achieve peak carbon and carbon neutrality targets, a high number of distributed power sources have been connected to distribution networks. How to realize the planning of a distribution network containing integrated energy under the condition of carbon capture and complete the exceedance test of the distribution network under the condition of accessing a large number of distributed generators has become an urgent problem. To solve the above problem while promoting sustainable development, this work proposes an active distribution network risk-planning model based on multisource data from carbon capture and the Power Internet of Things. The model calculates the semi-invariants of each order of the node state vectors and branch circuit current vectors and then utilizes Gram–Charlier-level expansion to obtain the exceeding probability density function and the probability distribution functions of the node voltages and line powers in the distribution network. Combined with multisource data, an active distribution network with an integrated energy system designed for carbon capture was modeled. According to the risk scenario of the distribution network, the nonconvex constraints in the model were simplified by second-order cone relaxation, and the optimal planning scheme of the distribution network was solved by combining the Gurobi solver with the risk index as the first-level objective and the economic benefit as the second-level objective. The simulation results of a coupled network consisting of a 39-node distribution network and an 11-node transportation network verified the effectiveness of the proposed model.