Azure blob trigger connection outside of Values

The value in local.settings.json will be take to environment variable. So if there is something need to get from environment variable, then you must set it in the Values. This is by design, azure function on local doesn't read settings from local.settings.json directly, it read settings from the environment variable.

From your situation, you have two ways to achieve what you want:

1, Let your settings like this:

  "Values": {
    "MyStorageSettings_ConnectionString": "myConnectionString",
    "MyStorageSettings_ContainerName": "myStorageContainerName",
    "MyStorageSettings_FileName": "filename.json"
  }

2, Create your own code to parsing json file.

using System;
using System.IO;
using System.Threading.Tasks;
using Microsoft.AspNetCore.Mvc;
using Microsoft.Azure.WebJobs;
using Microsoft.Azure.WebJobs.Extensions.Http;
using Microsoft.AspNetCore.Http;
using Microsoft.Extensions.Logging;
using Newtonsoft.Json;
using Newtonsoft.Json.Linq;
using System.Text;

namespace HttpTrigger
{
    public static class Function1
    {
        public static string GetFileJson(string filepath)
        {
            string json = string.Empty;
            using (FileStream fs = new FileStream(filepath, FileMode.Open, System.IO.FileAccess.Read, FileShare.ReadWrite))
            {
                using (StreamReader sr = new StreamReader(fs, Encoding.GetEncoding("utf-8")))
                {
                    json = sr.ReadToEnd().ToString();
                }
            }
            return json;
        }
        //Read Json Value
        public static string ReadJson()
        {
            string jsonfile = "custom.json";
            string jsonText = GetFileJson(jsonfile);
            JObject jsonObj = JObject.Parse(jsonText);
            string value = ((JObject)jsonObj["MyStorageSettings"])["ConnectionString"].ToString();
            return value;
        }
        [FunctionName("Function1")]
        public static async Task<IActionResult> Run(
            [HttpTrigger(AuthorizationLevel.Function, "get", "post", Route = null)] HttpRequest req,
            ILogger log)
        {
            string value = ReadJson();

            log.LogInformation("C# HTTP trigger function processed a request.");

            string name = req.Query["name"];

            string requestBody = await new StreamReader(req.Body).ReadToEndAsync();
            dynamic data = JsonConvert.DeserializeObject(requestBody);
            name = name ?? data?.name;

            return name != null
                ? (ActionResult)new OkObjectResult($"Hello, {name}")
                : new BadRequestObjectResult("Please pass a name on the query string or in the request body" + value);
        }
    }
}