Addressing the limitations of individual attribution scores via the Shapley value (SV), the field of explainable AI (XAI) has recently explored intricate interactions of features or data points. In particular, extensions of the SV, such as the Shapley Interaction Index (SII), have been proposed as a measure to still benefit from the axiomatic basis of the SV. However, similar to the SV, their exact computation remains computationally prohibitive. Hence, we propose with SVARM-IQ a sampling-based approach to efficiently approximate Shapley-based interaction indices of any order. SVARM-IQ can be applied to a broad class of interaction indices, including the SII, by leveraging a novel stratified representation. We provide non-asymptotic theoretical guarantees on its approximation quality and empirically demonstrate that SVARM-IQ achieves state-of-the-art estimation results in practical XAI scenarios on different model classes and application domains.