The Auto Update feature uses Indy's TIdHTTP. There's an effort to port Indy for Lazarus, but the Sourceforge file is a bit outdated (almost two years old). So I started to search for alternatives, just in case. I've found LNet (LightWeight Networking Library), written by Ales Katona. In the next days I'll try them all and decide which one the project will use.
Update: after some chatting on #lazarus-ide, I managed to download a newer version of indy4lazarus. I'll test it soon and post here the results.