edit: see the bottom, there is the real answer.

I encourage you yo use the Promise API. Your problem can be solved using a Promise.all call:

let promises = [];
while(something){
    promises.push(new Promise((r, j) => {
        YourAsyncCall(() => r());
    });
}
//Then this returns a promise that will resolve when ALL are so.
Promise.all(promises).then(() => {
    //All operations done
});

The syntax is in es6, here is the es5 equivalent (Promise API may be included externally):

var promises = [];
while(something){
    promises.push(new Promise(function(r, j){
        YourAsyncCall(function(){ r(); });
    });
}
//Then this returns a promise that will resolve when ALL are so.
Promise.all(promises).then(function(){
    //All operations done
});

You can also make your api call return the promise and push it directly to the promise array.

If you don't want to edit the api_call_method you can always wrap your code in a new promise and call the method resolve when it finishes.

edit: I have seen now the point of your code, sorry. I've just realized that Promise.all will not solve the problem.

You shall put what you posted (excluding the while loop and the control value) inside a function, and depending on the condition calling it again.

Then, all can be wraped inside a promise in order to make the external code aware of this asynchronous execution. I'll post some sample code later with my PC.

So the good answer

You can use a promise to control the flow of your application and use recursion instead of the while loop:

function asyncOp(resolve, reject) {
    //If you're using NodeJS you can use Es6 syntax:
    async_api_call("method.name", {}, (result) => {
      if(result.error()) {
          console.error(result.error());
          reject(result.error()); //You can reject the promise, this is optional.
      } else {
          //If your operation succeeds, resolve the promise and don't call again.
          if (result.data().length === 0) {
              asyncOp(resolve); //Try again
          } else {
              resolve(result); //Resolve the promise, pass the result.
          }
      }
   });
}

new Promise((r, j) => {
    asyncOp(r, j);
}).then((result) => {
    //This will call if your algorithm succeeds!
});

/*
 * Please note that "(...) => {}" equivals to "function(...){}"
 */

sigmasoldier's solution is correct, just wanted to share the ES6 version with async / await:

const asyncFunction = (t) => new Promise(resolve => setTimeout(resolve, t));

const getData = async (resolve, reject, count) => {

    console.log('waiting');
    await asyncFunction(3000);
    console.log('finshed waiting');

    count++;

    if (count < 2) {
        getData(resolve, reject, count);
    } else {
        return resolve();
    }
}

const runScript = async () => {
    await new Promise((r, j) => getData(r, j, 0));
    console.log('finished');
};

runScript();

If you don't want to use recursion you can change your while loop into a for of loop and use a generator function for maintaining done state. Here's a simple example where the for of loop will wait for the async function until we've had 5 iterations and then done is flipped to true. You should be able to update this concept to set your done variable to true when your webservice calls have buffered all of your data rows.

let done = false;
let count = 0;
const whileGenerator = function* () {
    while (!done) {
        yield count;
    }
};

const asyncFunction = async function(){
    await new Promise(resolve => { setTimeout(resolve); });
};
const main = new Promise(async (resolve)=>{
    for (let i of whileGenerator()){
       console.log(i);
       await asyncFunction();

       count++;
       if (count === 5){
           done = true;
       }
    }
    resolve();
});
main.then(()=>{
    console.log('all done!');
});

Also you may try recursion solution.

function asyncCall(cb) {
// Some async operation
}

function responseHandler(result) {
    if (result.error()) {
        console.error(result.error());
    } else if(result.data() && result.data().length) {
        asyncCall(responseHandler);
    }
}

asyncCall(responseHandler);

Here is a solution I came up with. Place this in an async function.

        let finished = false;
        const loop = async () => {
            return new Promise(async (resolve, reject) => {
                const inner = async () => {
                    if (!finished) {
                        //insert loop code here
                        if (xxx is done) { //insert this in your loop code after task is complete
                           finshed = true;
                           resolve();
                        } else {
                           return inner();
                        }
                    }
                }
                await inner();
            })
        }
        await loop();

If you don't want to use Promises you can restructure your code like so:

var tasks = [];
var index = 0;

function processNextTask()
{
    if(++index == tasks.length)
    {
        // no more tasks
        return;
    }

    async_api_call(
        "method.name", 
        { 
            // Do stuff.
        },
        function(result) 
        {
            if(result.error())
            {
                console.error(result.error());
            }
            else
            {
                // process data
                setTimeout(processNextTask);
            }
        }
    );
}

Your loop won't work, because it is sync, your async task is async, so the loop will finish before the async task can even respond. I'd reccomend you to use Promises to manage async tasks:

//first wrapping your API into a promise
var async_api_call_promise = function(methodName, someObject){
    return new Promise((resolve, reject) => {
        async_api_call(methodName, someObject, function(result){
            if(result.error()){ 
                reject( result.error() ) 
            }else{
                resolve( result.data() )
            }
        });
    })
}

now to your polling code:

//a local utility because I don't want to repeat myself
var poll = () => async_api_call_promise("method.name", {/*Do stuff.*/});

//your pulling operation
poll().then(
    data => data.length === 0 || poll(),  //true || tryAgain
    err => {
        console.error(err);
        return poll();
    }
).then((done) => {
    //done === true
    //here you put the code that has to wait for your "loop" to finish
});

Why Promises? Because they do state-management of async operations. Why implement that yourself?

  let taskPool = new Promise(function(resolve, reject) {
    resolve("Success!");
  });
  let that = this;
  while (index < this.totalPieces) {
    end = start + thisPartSize;
    if (end > filesize) {
      end = filesize;
      thisPartSize = filesize - start;
    }
    taskPool.then(() => {
      that.worker(start, end, index, thisPartSize);
    });
    index++;
    start = end;
  }

I've actually switched to undwinding a while loop into a for loop instead of my other answer to use a generator. It's a bit cleaner and more straight forward as to what is happening without the need for extra state.

const asyncFunction = async ()=>{
  await new Promise(resolve => { setTimeout(resolve); });
}

(async()=>{
  let i = 0;
  for (;;){
    await asyncFunction();
    
    console.log(i);
    // you have to manually perform your while condition in the loop
    if (i === 5) {
      // condition is true so break the infinite for loop
      break;
    }
    i++;
  }
  console.log('done');
})();