Firefox has an about:config setting called dom.event.clipboardevents.enabled, which allows Firefox to emit oncopy, oncut, and onpaste events which websites can use to modify the clipboard. You can set it to false to disable access to the Clipboard API across all websites; you can't use it to give websites access on a case-by-case basis. Setting this to false breaks some functionality such as pasting images on Discord.
Further, websites can use the deprecated Document.execCommand() copy and cut commands. Support for this can be disabled by setting the about:config preference dom.allow_cut_copy to false.
There is also the about:config setting dom.events.asyncClipboard.dataTransfer, which is disabled by default. If enabled, it grants websites to asynchronously read the clipboard (after receiving permission from the user), and to write to the clipboard at any time, without having to wait for an oncopy/oncut/onpaste event.