Dear Matthew Shuster,
Good day! Thank you for posting to Microsoft Community. We are happy to help you.
Per the description shared, I would like to summarize your concern i.e., you are not able to see embed code for the Microsoft Form.
If my understanding is correct, I would like to let you know that the person who created the Form in the Microsoft Forms page will be the default owner of the Form and when the creator click on the Collect Responses can see the Embed icon:

Since you mentioned the owner is not able to see the icon, can you please ask the owner to access the form in the different browser or InPrivate browser to see whether icon is present.
If not, can you see whether the problem is only with a specific form or every form created by the user?
We look forward to your response. Thanks for your cooperation.
Sincerely,
Mia | Microsoft Community Moderator