Machine learning has become an effective tool for automatically annotating unstructured data (e.g., images) with structured labels (e.g., object detections). As a result, a new programming paradigm called neurosymbolic programming has emerged where users write queries against these predicted annotations. However, due to the intrinsic fallibility of machine learning models, these programs currently lack any notion of correctness. In many domains, users may want some kind of conservative guarantee that the results of their queries contain all possibly relevant instances. Conformal prediction has emerged as a promising strategy for quantifying uncertainty in machine learning by modifying models to predict sets of labels instead of individual labels; it provides a probabilistic guarantee that the prediction set contains the true label with high probability. We propose a novel framework for adapting conformal prediction to neurosymbolic programs; our strategy is to represent prediction sets as abstract values in some abstract domain, and then to use abstract interpretation to propagate prediction sets through the program. Our strategy satisfies three key desiderata: (i) correctness (i.e., the program outputs a prediction set that contains the true output with high probability), (ii) compositionality (i.e., we can quantify uncertainty separately for different modules and then compose them together), and (iii) structured values (i.e., we can provide uncertainty quantification for structured values such as lists). When the full program is available ahead-of-time, we propose an optimization that incorporates conformal prediction at intermediate program points to reduce imprecision in abstract interpretation. We evaluate our approach on programs that take MNIST and MS-COCO images as input, demonstrating that it produces reasonably sized prediction sets while satisfying a coverage guarantee.