Hi Jenny,
Check out this blog post about troubleshooting embedded apps and websites. It looks like your company's website falls under the category at the bottom -- "Some websites restrict embedding". When I try to embed that site, I get this error in Chrome Developer Tools:
Refused to display 'https://www.prowestgis.com/' in a frame because it set 'X-Frame-Options' to 'sameorigin'.
Since it's your own company's website, your IT staff may have some insight on how to allow embedding of your website in a Story Map.
(Many major companies also limit embedding of their websites, as you found with Google and Twitter).
I also found this web app that could help with checking the embeddability of sites:
Free iframe checker | TinyWebGallery Blog