A game engine that exports to HTML5 is going to create an html that runs the game. (This is the same index.html file that itch.io expects for HTML5 embeds.) You can edit that page to add the frame around your game's viewport. The easiest way might be to use background CSS positioned correctly. Afterwards, if you upload to itch.io you'll need to make sure you set the dimensions of your game's viewport large enough.
Another approach would be to have your game render the frame itself. This might be easier for you if you're more familiar with your game engine.