This is a plugin that integrates Antidote with Obsidian. Simply install and the plugin, and then use the "Correct with Antidote" command to open the selected text in Antidote.
This code is largely based on Druide's sample Visual Studio Code for integration with the Connectix Agent. In fact, the Antidote/ folder is copied from it, except for the VSCode APIs, which where replaced with their closest Obsidian equivalent.