I have seen many different approaches. It really depends on the country, how EPI is organized and who has the responsibility to monitor service delivery and the supply chain. In some countries, the EPI is reposnible for both, in others the EPI for one or the other, or none; in some countries the supply chain is monitored by a LMIS that includes not only vaccines but other medications, in others not; same with HMIS, in some countries this is the only system to monitor doses administered, while in other countries there are parallel systems EPI and HMIS.
The most important aspect, I would say, is to ensure that whatever electronic info system is used meets all the needs (managerial and for monitoring) and functional standards requiered by all persons involved, at different levels. In a few countries (that I have seen), they have managed to integrate both systems and make them inteact with each other. For example, in teh city of Bogota, there is an electronic registry that crosses info with the supplies used and wastage.
I would recommend reading the Optimize Manual on the topic: "Planning an Information Systems Project: A Toolkit for Public Health Managers". This toolkit can help public health managers plan for the implementation of information and communications technology in health information systems. It draws on lessons learned by project Optimize, a five-year collaboration between the World Health Organization and PATH, to help optimize the vaccine supply chain. Author(s): Grevendonk J, Taliesin B, Brigden D. Corporate author(s): PATH, World Health Organization. Publication date: July 2013. Available at: https://www.path.org/publications/detail.php?i=2343