I think there are a few options:
(1) Use the aDosBox app and run EpiInfo for DOS as your data-entry system. The resulting .REC files can be used with EpiInfo for Windows. Note that EpiNut in EpiInfo for DOS provided only the older NCHS reference. I have found that I can run EpiInfo for DOS using DosBox on MAC OS-X and iPad/IPod/IPhone iOS. It should work on Android devices.
(2) The DosBox Turbo can run Windows 95, 98, possibly ME (all of which were DOS-based - it will not run Windows NT, 2000, XP, Vispa, 7, or 8 as these are not DOS-based products). You could try running Windows 98 and installing / running ENA (check tat ENA will run on Windows 98). That would give you ENA on Android. I have no experience with this but your IT people should be able to get this to work.
(3) Similar to above ... use the Bochs "virtual PC" app on Android. This will run any version fo Windows. Videos available here and here. This is probably too slow for real use.
(4) Your Android data system should be able to export data to text files. If it can do this with comma or tabs separating values then you can import this easily into Excel and save in Excel format for use in ENA.
I hope this is of some use.