Stay-at-home orders (SAHOs) have been implemented in most U.S. states to mitigate the spread of COVID-19. This paper quantifies the short-run impact of these containment policies on the supply of and demand for child care. The child care market may be particularly vulnerable to a SAHO-type policy shock, given that many providers are liquidity-constrained. Using plausibly exogenous variation from the staggered adoption of SAHOs across states, we find that online job postings for early care and education teachers declined by 13% after enactment. This effect is driven exclusively by private-sector services. Indeed, hiring by public programs like Head Start and pre-kindergarten has not been influenced by SAHOs. In addition, we find little evidence that child care search behavior among households has been altered. Because forced supply-side changes appear to be at play, our results suggest that households may not be well-equipped to insure against the rapid transition to the production of child care. We discuss the implications of these results for child development and parental employment decisions.