Google-sheets – Increment number in array formula based on a condition

I've added the result of your question in the Google Spreadsheet you shared with us.
In the SUMMARY sheet I collected all data that fits the criterion D<>0, with the following formula:

=QUERY(DATA!A:D;"SELECT A,C,D WHERE D<>0")

I also added an ID number, for the two companies and added the column to the summary sheet. As you can see, this formula can be easily extended.

If you did not insist on array-based formula, then

sumifs(C:C, A:A, A2, B:B, "<="&B2)

entered in D2 and copied around the D column, would do the job.

I don't think that an arrayformula-based solution exists, because arrayformula runs through an array once, performing some element-wise operations. You want to compare each element with all others, which is one step further in complexity (a double for loop).

But you can use a custom function, something like =runningtotal(A2:A, B2:B, C2:C) which processes the columns and returns an array of running totals. The script implementing this function is below; add it via Tools > Script Editor. The logic is quite straightforward: as said above, it's a double for loop.

function runningTotal(names, dates, amounts) {
  var sum, totals = [], n = names.length;
  if (dates.length != n || amounts.length != n) {
    return 'Error: need three columns of equal length'; 
  }
  for (var i = 0; i < n; i++) {
    if (names[i][0]) {
      sum = 0;
      for (var j = 0; j < n; j++) {
        if (names[j][0] == names[i][0] && dates[j][0] <= dates[i][0]) {
          sum = sum + amounts[j][0];
        }
      }
    }
    else {
      sum = '';
    }
    totals.push([sum]);
  }
  return totals;
}