Starting from one totipotent cell, complex multicellular organisms form through a series of differentiation and morphogenetic events, culminating in a multitude of cell types arranged in a functional and intricate spatial pattern. To do so, cells coordinate with each other, resulting in dynamics which follow a precise developmental trajectory, constraining the space of possible embryo-to-embryo variation. Using recent single-cell sequencing data of early ascidian embryos, we leverage natural variation together with modeling and inference techniques from statistical physics to investigate development at the level of a complete interconnected embryo - an embryonic transcriptome. After developing a robust and biophysically motivated approach to identifying distinct transcriptomic states or cell types, a statistical analysis reveals correlations within embryos and across cell types demonstrating the presence of collective variation. From these intra-embryo correlations, we infer minimal networks of cell-cell interactions, which reveal the collective modes of gene expression. Our work demonstrates how the existence and nature of spatial interactions along with the collective modes of expression that they give rise to can be inferred from single-cell gene expression measurements, opening up a wider range of biological questions that can be addressed using sequencing-based modalities.