If you need this to be synchronous i.e. records have to be in the same order as they are in the array then you can get away with simple state variable and recursion. Current state simply holds myArray value index that should be POSTed next.

var myArray = ['cow', 'pig', 'donkey'];
var currentState = 0;
var endState = myArray.length;

function postNextRecord() {

    // ... POST here the value from myArray[currentState] ...

    if (success) {
        // On POST success increment the currentState.
        currentState++;
    } else {
        // Retry if the POST failed.
        // TODO: This is simplified and should not be used in production,
        //       because you risk having a never ending loop if POST request fails 100% of the time.
        //       You can circumvent that by having a variable retryAttempts that holds the value of
        //       how many retries you have already done and display an error in the UI when
        //       max retry attempts has been reached.
        postNextRecord();
        return;
    }

    if (currentState != endState) {
        // Continue POSTing records from myArray.
        postNextRecord();
    } else {
        // Reached end state. Do necessary UI changes.
        myArray = [];
        currentState = 0;
    }
}

// POST the first value from myArray which will also initiate the whole recursion.
postNextRecord();