I'm not sure about Discord being screen reader accessible, but the closest thing I can think of to that at the moment, is the "/tts" or "text-to-speech" command that can be activated. It will read what you've posted to everyone in the channel or chat room (if they have it activated). I'm not certain if the default setting has it turned on, but it is in the User Settings, under Text & Images, and is the option all the way at the bottom.
Also, if you turn on Text-To-Speech Notifications for all channels, if someone sends a message (that is text), the voice will also read everything they post to you and/or that channel you're in (this is under Notifications).
The only problem is that I think it may be a one time reading of the text that has been inputted, and am not certain if there's a way to get it to replay it.