There's a whole ecosystem for that sort of thing, but you have to know where to look. It depends on what flavor of reading you want; a polished, scripted performance or a more raw, live-chat-with-the-author vibe. Platforms like YouTube and Twitch are your starting points. A lot of authors will post prerecorded audio readings of their chapters, sometimes with music or sound effects, and call them 'live readings' even if they aren't streamed.
For truly live, interactive stuff, I've seen authors use Discord stages or Twitch streams to read new chapters aloud to their patrons or followers, taking requests for voices or pausing for reactions. It feels more like a community event. The real trick is finding authors who do this, because it's not super common. You'll have the best luck by following your favorite fanfic writers on social media and seeing if they ever announce something like a 'chapter launch party' on a voice channel.
I stumbled into one for a popular 'Undertale' ship fic last year, and it was surprisingly fun. The author was nervous but the chat was super supportive, and they did different voices for Sans and the other characters. It wasn't professional voice acting, but that made it charming. Since then, I keep an eye on the 'live' tags on Tumblr and the events sections in fandom-specific Discord servers.