For people wanting to use the built-in .NET SmtpClient rather than the SendGrid client library (not sure if that was the OP's intent), I couldn't get it to work unless I used apikey
as my username and the api key itself as the password as outlined here.
<mailSettings>
<smtp>
<network host="smtp.sendgrid.net" port="587" userName="apikey" password="<your key goes here>" />
</smtp>
</mailSettings>