Wininet is part of IE5. If you've got the IE5.5 beta version provided with your WinME, then go to the microsoft site and download the latest version. If you have the most up to date version, then try repairing your current installation by going to control panel, add/remove programs and selecting IE5.5 then the repair option.
I have a similar error message, but it won't let me do anything than OK it and restart. I cannot get into any programs, and only the Start bar is on the screen, which does not react to anything. I'm trying to get a DOS prompt so I can delete the temporary internet files and history folder, as I've heard this can fix it, but I can't even get into DOS!!! Any help greatly appreciated!
I managed to fix this problem myself in the end and will now share my voyage.... I renamed wininet.dll to wininet.old and it allowed me further into the loading of Windows ME. The error message now was about kernel32.dll. After pressing close, I was allowed to load my CD burner software and back everything up. I re-installed Windows ME but the problem was still there. So, with all files backed up, I re-formatted the C drive, re-installed it and, of course, all is now fine. I know it sounds a bit drastic, but it does work. Apparently, if you partition your hard drive and put all your personal files on the newly created D drive, you can happily format the C drive and keep all your files...
