We wish! Currently, the API is supported on Chrome 86+, Edge 86+ and Opera 72+: https://developer.mozilla.org/en-US/docs/Web/API/File_System.... It's all quite new, and we hope other browsers will follow.
[1] With the Webextensions API, you have to decide on an action before starting the download, and the only choices are "overwrite", "rename the new download" or "prompt the user". There's no "skip the download" option for example, and Firefox doesn't even support the "prompt the user" option, either.