First - it's 'Ordnance' - the term 'ordinance' implies some variety of ruling...
That said - the Ordnance Officer of the time was probably Back East in a comfortable Arsenal or Post - working on things like R&D - maintaining and updating the Army's various equipments dealing with weapons.
The Officer making the request was the Commander of whatever Post that needed ammunition, and was determined at Post/Camp level on a yearly basis - based upon what was still on hand from what the Army had supplied him with from the previous year.
It was then forwarded through the Department overseeing his area of operations and on to the War Department - who decided if a penurious Congress had allocated enough money to warrant 'all' of the proposed expenditure.
Often - it hadn't.
This accounts for the dearth of 'practice ammunition' and the subsequent decline in marksmanship practice.
Getting ammunition shipped to the various locations held by the Frontier Army was handled through the Quartermaster's contract teamsters, who handled the freighting, and it wasn't fast, by any means, since the ammunition had to be on-hand at the Depot in order for it to be shipped.
If it wasn't - it came from the Arsenal - usually via Rail.
Prior to 1873 - ammunition went out to those units who were issued the particular weapon, and with remarkably little confusion, but then the requests were pretty specific, and hand-written neatly.
After 1873 - the standardization of the Army's small arms and ammuntion aided greatly in its distribution as the new weapons made their way into issue.
That system only improved over time - but it certainly wasn't like today's military who can forecast ammunition expentiture for peacetime training purposes at least three years out - and even that is dependent upon allocation of monies - with many units only firing for 'Familiarization' - as opposed to 'Qualification' - every other year, if that.
Unless an individual Commander with experience and foresight had stockpiled ammunition in some way - quick re-supply or an emergency issue was pretty much out of the question, and the modern concept of long-range planning didn't seem to hold quite as much water in the rapidly-changing West, due to the continual movement of 'front lines'.
Damned difficult to plan against a fluid force with no fortifications to defend, you see.
Vaya,
Scouts Out!