Distributed lags play important roles in explaining the short-run dynamic and long-run cumulative effects of features on a response variable. Unlike the usual lag length selection, important lags with significant weights are selected in a distributed lag model (DLM). Inspired by the importance of distributed lags, this research focuses on the construction of distributed lag inspired machine learning (DLIML) for predicting vaccine-induced changes in COVID-19 hospitalization and intensive care unit (ICU) admission rates. Importance of a lagged feature in DLM is examined by hypothesis testing and a subset of important features are selected by evaluating an information criterion. Akin to the DLM, we demonstrate the selection of distributed lags in machine learning by evaluating importance scores and objective functions. Finally, we apply the DLIML with supervised learning for forecasting daily changes in COVID-19 hospitalization and ICU admission rates in United Kingdom (UK) and United States of America (USA). A sharp decline in hospitalization and ICU admission rates are observed when around 40% people are vaccinated. For one percent more vaccination, daily changes in hospitalization and ICU admission rates are expected to reduce by 4.05 and 0.74 per million after 14 days in UK, and 5.98 and 1.04 per million after 20 days in USA, respectively. Long-run cumulative effects in the DLM demonstrate that the daily changes in hospitalization and ICU admission rates are expected to jitter around the zero line in a long-run. Application of the DLIML selects fewer lagged features but provides qualitatively better forecasting outcome for data-driven healthcare service planning.