Blobs are "immutable" so you can't change one after making it. Constructing a new Blob that appends the data to an existing blob (as you wrote in your initial question) is a good solution. 
If you don't need to use the Blob each time you append a part, you can just track an array of parts.  Then you can continually append to the array and then construct the Blob at the end when you need it.
var MyBlobBuilder = function() {
  this.parts = [];
}
MyBlobBuilder.prototype.append = function(part) {
  this.parts.push(part);
  this.blob = undefined; // Invalidate the blob
};
MyBlobBuilder.prototype.getBlob = function() {
  if (!this.blob) {
    this.blob = new Blob(this.parts, { type: "text/plain" });
  }
  return this.blob;
};
var myBlobBuilder = new MyBlobBuilder();
myBlobBuilder.append("Hello world, 2");
// Other stuff ... 
myBlobBuilder.append(",another data");
var bb = myBlobBuilder.getBlob();