Caniuse?
At the time of this writing, browser support for the dialog element is about 70% globally. But there is hope for more. Firefox's implementation is behind a flag. Edge status for the dialog is under consideration, stating "...likely for a future release." You can cast votes supporting development on Edge.
Polyfill
Now you need a polyfill. Audible sigh. That being said, I believe it's worthwhile to continue with dialog, even in this intermediate state. If you're with me, here are two options to consider right now.
Google Chrome Polyfill
Pros: drop-in polyfill from the Google Chrome team. Easy to use. Cons: not super small.
https://github.com/GoogleChrome/dialog-polyfill
a11y-dialog
Pros: very small. Built with accessibility in mind. React and Vue variants available. Cons: requires more markup than a drop-in polyfill.
https://github.com/edenspiekermann/a11y-dialog
React: https://github.com/HugoGiraudel/react-a11y-dialog
Vue: https://github.com/morkro/vue-a11y-dialog