Hi Megan,
I apologize for the trouble. Assuming your emails are sending from Express Invoice properly and reaching their intended destinations (it sounds like they are), then I believe this is an issue on Yahoo's side.
A few things to try:
1. There may be a setting in Yahoo's email that you can change that will allow it to retain sent emails in the way you're looking for. I would try telling their support that you are using SMTP to send emails from a 3rd party client, and they send OK, but they don't appear in your Sent Mail box, and see if they have any ideas.
2. Kind of a workaround, but you can probably create a rule in your Yahoo Inbox to move the emails to your Sent Items folder where you actually want them.