Legal intake, the process of finding out if an applicant is eligible for help from a free legal aid program, takes significant time and resources. In part this is because eligibility criteria are nuanced, open-textured, and require frequent revision as grants start and end. In this paper, we investigate the use of large language models (LLMs) to reduce this burden. We describe a digital intake platform that combines logical rules with LLMs to offer eligibility recommendations, and we evaluate the ability of 8 different LLMs to perform this task. We find promising results for this approach to help close the access to justice gap, with the best model reaching an F1 score of .82, while minimizing false negatives.