Explain what? What a service account is? You log on to the machine which runs SQL Server and IIS as DOMAIN\Admin or whatever. You run the Always Encrypted wizard, and the certificate ends up in the certificate store of DOMAIN\Admin.
However, the web server does typically not run as DOMAIN\Admin, but some other account. I have feeling that IIS typically as as Local Service, but I know very little about IIS, and you may be better off asking in a forum for IIS (if that is the web server you are using) for details. I googled around, and I found https://stackoverflow.com/questions/38462367/where-does-one-place-the-always-encrypted-certificate-on-an-iis-7-5-web-server. There are two solutions suggested. The first one may be the simpler one, but I don't like it from a security standpoint, so I would suggest that you try the second.
I don't know if you have made your initial tests with IIS Express, but this is different. IIS Express runs in your user space, so in this case, it will find your personal certificate store.