Unfortunately, as the folder structure would suggest (/sitecore/admin) the users need to be full Administrators.
The CheckSecurity() method of the UpdateInstallationWizard base class specifically checks for this:
private bool CheckSecurity()
{
User user = Context.User;
if ((Account) user != (Account) null && user.IsAdministrator)
return true;
SiteContext site = Context.Site;
string url = (site != null ? site.LoginPage : string.Empty) + "?returnUrl=" + this.Server.UrlEncode(string.Format("{0}.aspx{1}", (object) this.PageUrl, string.IsNullOrEmpty(this.Request.QueryString.ToString()) ? (object) string.Empty : (object) ("?" + this.Request.QueryString.ToString())));
if (url.Length > 0)
this.Response.Redirect(url, true);
return false;
}
You can check this yourself in dotPeek decompiler. It's defined in class Sitecore.Update.UpdateInstallationBase
in the Sitecore.Update dll