How would one acquire an orc army?

[Legate of Amon Lanc]

I think the main clue for start at least lies in the fact that Saruman's home was basically next to Misty Mountains. He could have sent his servants to capture some random Orcs, or bargain with one or two minor tribes to come to serve him. He could surely lure them with some small amount of e.g. precious metals (which he surely had access to) or promises of spoils of war or whatever. Even putting aside the fact that he was a master diplomat, probably it is easy to convince a tribe whose daily reality is fighting cold, not very much food and generally bleak existence in the mountains.

As for hosting the Orcs and hiding them from the Wise, we are told that throughout the centuries, Saruman became more and more secretive and locked himself up in Isengard, which became rather isolated from the rest of the world. Note that we don't know e.g. that the White Council would ever have been hosted in Isengard. The visits of the Wise were scarce, maybe one Radagast per half a century, if there was need of him. And maybe not even that.
It is clear from the account in FotR that Gandalf had not visited Isengard for years before his coming and subsequent imprisonment there (in fact, it seems they perhaps even had not seen each other since the Dol Guldur assault in the year Bilbo found the Ring). The Council members did not seem to be so much in contact with each other. Elven Lords were mostly sitting in their own havens, Radagast messing around in the Wilderness, Gandalf kept hurrying here and there after his own business. I have no reason to think that other random Elves etc. (for example some messengers from Elrond) would have visited Saruman either. It seems to me there was a general air of laziness and the feeling of unnecessarity to communicate among the Wise as the Third Age grew older.

So Saruman had only to hide the Orcs from a "casual viewer", i.e. make sure there weren't ten thousand Orcs randomly walking through Isengard in plain daylight. I believe they were all stationed underground and all over the place, and the description of Isengard makes it clear it was exactly suitable for that:

Originally Posted by Two Towers, Chapter 8: The Road to Isengard

Many houses there were, chambers, halls, and passages, cut and tunnelled back into the walls upon their inner side, so that all the open circle was overlooked by countless windows and dark doors. Thousands could dwell there, workers, servants, slaves, and warriors with great store of arms; wolves were fed and stabled in deep dens beneath. The plain, too, was bored and delved. Shafts were driven deep into the ground; their upper ends were covered by low mounds and domes of stone, so that in the moonlight the Ring of Isengard looked like a graveyard of unquiet dead. For the ground trembled. The shafts ran down by many slopes and spiral stairs to caverns far under; there Saruman had treasuries, store-houses, armouries, smithies, and great furnaces.

I think it is clear there was no problem literally hiding thousands of Orcs somewhere at (or under) Isengard.

The size of Isengard's army and how it was acquired in relatively quick manner is probably the most interesting thing, but then, Saruman probably really took the "good old breeding program guidelines" from Morgoth or whatever, after all, he had the deep knowledge of the "arts of the Enemy". When you read about his machines etc., he is described as being very efficient in making things very efficient, so I have no doubt he was able to breed a lot of Orcs over a couple of decades (unlike the couple of months like the movies would have us believe).