Create a blob storage container programmatically

CloudBlobContainer container = new CloudBlobContainer("https://mystore.blob.core.windows.net/" + subDomain);

This just creates an instance of CloudBlobContainer object. To create a container in your storage account, you would need to call CreateIfNotExists or Create function on this object.

Try this code instead:

public void AddCompanyStorage(string subDomain)
    {
        //get the storage account.
        CloudStorageAccount storageAccount = CloudStorageAccount.Parse(
            System.Configuration.ConfigurationManager.AppSettings["StorageConnectionString"].ToString());

        //blob client now
        CloudBlobClient blobClient = storageAccount.CreateCloudBlobClient();            

        //the container for this is companystyles
        CloudBlobContainer container = blobClient.GetContainerReference(subDomain);
        container.CreateIfNotExists();

    }

    public void AddCompanyStorage(string subDomain)
        {
            //get the storage account.
            CloudStorageAccount storageAccount = CloudStorageAccount.Parse(
                System.Configuration.ConfigurationManager.AppSettings["StorageConnectionString"].ToString());

            //blob client now
            CloudBlobClient blobClient = storageAccount.CreateCloudBlobClient();            

            //the container for this is companystyles
           CloudBlobContainer container = blobClient.GetContainerReference(subDomain);

            //Create a new container, if it does not exist
           container.CreateIfNotExists();
        } 

Follow the guidline for container names:

• A container name must be a valid DNS name, conforming to the
  following naming rules: Container names must start with a letter or
  number, and can contain only letters, numbers, and the dash (-)
  character.

• Every dash (-) character must be immediately preceded and followed by a letter or number; consecutive dashes are not permitted in container names.

• All letters in a container name must be lowercase.

• Container names must be from 3 through 63 characters long.