Hello, erichollywoodfrodo!
Thanks for reachig out to the Apple Support Communities today about accessing Messages through a web browser. At this time, it's not possible to access Messages on the web. You can learn more about accessing your iMessages in this article to see What is the difference between iMessage and SMS/MMS?
"iMessages are texts, photos, or videos that you send to another iPhone, iPad, iPod touch, or Mac over Wi-Fi or cellular-data networks. These messages are always encrypted and appear in blue text bubbles. To turn iMessage on or off, go to Settings > Messages."
You're welcome to check out this link for Product Feedback to submit feedback directly to Apple for features you'd like to see in the future.
Have a wonderful day!