This is going to be a long one, ladies and gents.
So, here is the situation as I see it.
Ulfric controls Skyrim as High King, with Galmar Stone-Fist at his left hand and Stormblade, or the Dovahkiin, at his right. Already you’ve got a pretty good set-up for Ulfric there, the Dragonborn is a pretty much unstoppable force, and he or she (for the purpose of this post I’ll say he’s a male) is the most important factor here.
First, we need to take into account that in Lore, the Dragonborn would have been unstoppable. This is a video game, which is the reason that you can’t use endless shouts at a consistent rate, or combine shouts. But if we take those limitations out of the equation, he’s pretty damn unstoppable.
Next, we need to take into account the Dragon Breaks. A Dragon Break, for those who don’t know, is when multiple things happen in the universe, multiple timelines are created, all centred around one specific point, and then only one of those ends up being the correct version of events. For instance, there are around seven or so endings for Daggerfall; but which one really happened? A Dragon Break fixes that problem by deciding which one.
Here are some instances of Dragon Breaks in Skyrim: Which side did the Dovahkiin join? Of course, we’re assuming the Stormcloaks. Did the Dovahkiin destroy the Dark Brotherhood or join them and ultimately assassinate the Emperor? I’m going to assume the latter in this case. Was Paarthurnax killed? We’ll assume no. And there are many more, the most common of which is whenever your character dies and you reload a save. Did the Dragonborn die then, or did he not?
Now then. With the Dragonborn at his side, Ulfric’s pretty unstoppable, and not just because he’s now got the power of two Thu’um-users. There’s more.
As it stands, the Dragonborn has control of: The Dark Brotherhood, the Thieves Guild, the College of Winterhold, the Companions, is a friend to the Greybeards (however they probably won’t fight for him when asked) and even the Bard’s College. He also has the loyalty of Odahviing and at least one warrior of Sovngarde. All will do whatever they are told by him.
So here’s how each of these guys would be useful. The Dark Brotherhood are of course assassins, and would be very useful to the Stormcloaks in that they can slip in, kill a Thalmor general, and slip out without being noticed. The Thieves Guild could steal Thalmor documents. The College is admittedly slightly looser and the Archmage would not necessarily be able to get them all fighting, but he’d be able to convince at least some of them (J’zargo, for instance) to fight. Obviously mages would be very useful in the battlefield. The Companions would be useful for obvious reasons; they have some of the best warriors in Skyrim and two werewolves (Aela and the Dragonborn himself). The Bard’s College wouldn’t be that useful, except that they could keep up the morale of the troops and have some warriors in their ranks. As for Odahviing, well, that doesn’t need explaining.
The Greybeards present a greater problem. They refuse to involve themselves with the affairs of others - hell, they were hesitant to even get involved in peace talks! However, as someone else said, try attacking one and see how peace-loving they are. They would absolutely get involved if attacked by the Thalmor, giving Ulfric a total of seven Thu’um-users - the Dragonborn, Ulfric himself, the four Greybeards, and Odahviing - against the Thalmor’s zero. Now that is a hell of an advantage.
Next we examine Skyrim’s climate. The Dominion has very little presence in Skyrim, and don’t know the area particularly well, which already gives the Stormcloaks an advantage. Skyrim is also the perfect place for Guerrilla Combat, as is Hammerfell, which was the main reason that the Dominion was kicked out of there. Finally, in case you haven’t noticed, Skyrim’s damned cold. The Nords are used to cold, while the High Elves aren’t, leaving the Thalmor soldiers at another disadvantage.
However, here’s were things go South for the Stormcloaks. If, and we’re assuming he did, the Dragonborn did assist the Dark Brotherhood, then the Emperor was killed in Skyrim. Considering the war going on at the time, the Empire is going to blame the Stormcloaks for this (either thinking it was them or that they hired the Brotherhood to do it for them) so we can count out their help, even if Ulfric would accept it, which I doubt he would.
Next up we have the problem of the Elves in Windhelm. In the event of an invasion, I think we can assume that at least most of them would side with the Thalmor. Considering that Windhelm is the new capital of Skyrim, this would be a problem. However, with the combined efforts of Ulfric, the Dovahkiin, the Windhelm guard and the more warlike (read: most of the) citizens of Windhelm, this wouldn’t pose much of a threat.
So, weighing up the advantages and disadvantages of each side, as long as Ulfric keeps his men in Skyrim and lets the Dominion come to him, the Thalmor are royally fluffed.