Avoid using Request(), use Request.QueryString() or Request.Form() as needed. Also, make sure you have the AddRecipient parameters in the correct order, most email COMs use the following syntax:
Mailer.AddRecipient( strEmail, strName )
Try this code and see what it does for you (assuming your mailer is using the form POST method).
<sidenote>In the code you posted, you are actually sending
Set Mailer = Server.CreateObject( "SMTPsvg.Mailer" )
Mailer.FromName = Request.Form( "Name" )
Mailer.FromAddress = Request.Form( "Email" )
Mailer.AddRecipient "Upper Village Market Customer Order", "firstname.lastname@example.org"
Mailer.RemoteHost = "mail.webrothers.com"
Mailer.ReturnReceipt = False
Mailer.ConfirmRead = False
Mailer.TimeOut = 30
Mailer.Subject = "** Customer Order **"
Response.Write( Mailer.SendMail )
the email twice, as you call Mailer.SendMail 2 times, once
to send and once to check for errors.</sidenote>