Speech emotion recognition (SER) models typically rely on costly human-labeled data for training, making scaling methods to large speech datasets and nuanced emotion taxonomies difficult. We present LanSER, a method that enables the use of unlabeled data by inferring weak emotion labels via pre-trained large language models through weakly-supervised learning. For inferring weak labels constrained to a taxonomy, we use a textual entailment approach that selects an emotion label with the highest entailment score for a speech transcript extracted via automatic speech recognition. Our experimental results show that models pre-trained on large datasets with this weak supervision outperform other baseline models on standard SER datasets when fine-tuned, and show improved label efficiency. Despite being pre-trained on labels derived only from text, we show that the resulting representations appear to model the prosodic content of speech.