I've encountered the Goblin Master a couple of times.
The first time was when I caught her in a cage trap. I was interested and went into Legends mode and looked her up. In her life she had gone to the deeps and caught and tamed a whole lot of beasts. After the particular siege where she participated, and I killed a few dozen goblins and their mounts and so forth, no beasts ever showed up again. Neither did really big sieges. But I'd already killed at least hundreds, if not into four figures, of goblins by this point. It could have been that that last siege was a last-ditch effort of nearly everything they had left and when I killed them all, they were out of warriors.
The second time I killed the Master on the field of battle. None of the sieges up to that point had had mounts or war beasts, but I didn't really think too much about it so never looked in Legends mode to see anything about him. Subsequent sieges were still pretty big, although there were still no mounts or beasts. I killed him pretty early on, though, so it may just be that I hadn't drained the local civ of military goblins yet.
So I know this is a small sample size for !!SCIENCE!! but I've developed two hypothesis:
1) If the Master of the local civ never trained beasts, there won't be any in sieges. If you capture/kill him early enough, you'll never see mounts/beasts.
2) You can drain the pool of available goblins, if you kill enough of them.