By measuring the time delay between any pair of gravitationally lensed images, we can constrain the expansion rate of the universe and test dark energy models. Variable quasars have been used in this way with great success, and it is now possible to extend this technique to gravitationally lensed supernovae (SNe). These targets are especially promising because their predictable light curves can deliver precise time delay measurements in a relatively short period. Existing surveys have a realistic chance of discovering at least one new lensed SN within the next 1-3 years. This long-term target of opportunity program will provide the high-resolution follow-up imaging that is critical for measuring a lensed SN time delay. This program will enable a time delay measurement with uncertainty of approximately +-3 days, leading to a new measurement of H0 with a precision of 5-10% for a flat LambdaCDM cosmology---comparable to the best constraints achieved with lensed quasars. Future wide-field surveys (e.g., LSST, WFIRST) could deliver hundreds of lensed SN time delays, but the sample size will be limited by follow-up resources. This pilot study is an important step to establish lensed SNe as a competitive cosmological tool in the coming wide-field survey era.