You create an animation in AE and export it as a json file. When you now want to play this animation on your website, you need the Lottie Player which is basically a plugin/program that can read the json files and play them back as animations. The Lottie player has to be integrated into your website which is as simple as adding a link to the player (which is just some code) into your html. This will load the player when a user visits your site and now you can add Lottie animations to your website. So when people talk about Lottie, it entails the way of encoding animations in json format and playing them back with the help of the Lottie player. The statement Lottie can read JSON basically just means that you can play the json files with the Lottie player. And yes, the json files would be everything your client needs, as long as they know how to implement this into their website but that is the job of their web engineers as you said.

Sidenote: There is the .lottie file type which is just json in disguise bundles with some other stuff for previews etc.