Json in Web.Config AppSettings
Not nicely. The web.config is XML and quotes are not allowed in XML without escaping them to ". Doing that on a JSON still will make it "difficult" to read / maintain. Base64 that serhiyb is one solution, but that will not be maintainable by a person either.
I'd store them as appsettings key/value pairs and then generate the json string at runtime.
If you are 100% the string will never change, then escaping or Base64 is an option.
With your limitations I would just encode it to Base64 string and decode it back when you read it from web.config.
If you use " instead of quote characters then you can store JSON in app settings or connection strings without issues in the Web.config. E.g.:
<add key="ThisIsJson" value="[{"Name":"xyz"}]"/>This doesn't need any special processing either, at least with Json.NET.