While there is an element of ethnonationalism to the Stormcloaks, i dont remember having seen examples of Stormcloaks actually implementing racist policies or racist demeanor. They seem to be focused on having sovereignty over their territory (the people of Skyrim should make choices for themselves, not merely obey Cyrodill).

The main scene that convince people that the Stormcloaks are racists is the opening scene when you enter Windhelm. But people usually dont realize that:

1- the two people haranging the Dunmer are not Stormcloak soldiers themselves. They are a beggar and a drunk.
2- the main grievance they throw at the Dunmer is that "they dont join the Stormcloak". For an alleged Nord supremacist faction they are would be happy to have Dunmer join them
3- the situation of the Grey Quarter is a Windhelm problem, not a "Stormcloak" problem. The city accepted a large, large quantities of Dunmer refugees when their nation where in a time of crisis, two hundred years ago and they have stayed since.

So when the Dunmer replies that the civil war which is about Skyrim's self-determination, "is not their fight", i think they are making a statement that they owe nothing to the people who welcomed them to their home and they feel no attachment to the land that welcomed them.

(*Also lets not forget that Morrowind Dumners are xenophobic, arrogant, manipulators and Daedra worshippers. And slavers. It's their culture, but they arent the nicest people in the block.).

Finally, if the justification for the Dumner situation in Windhelm was purely racist, then why does Windhelm have the largest Altmer diaspora in all of Skyrim. There are more Altmer NPCs in Windhelm than anywhere else in the entire game world, but sure let's assume their entire faction/city is just Nord Supremacist.

Also remember that the Argonian ban in Windhelm is because there is so much bad blood between Argonians and Dunmer, because the Dunmer enslaved the Argonians for millenia. Even if the city gets under Imperial rule, the ban doesn't get lifted FOR THE ARGONIAN OWN PROTECTION.

Not to say there aren't some terribly racist sentiments against Dunmers existing in Windhelm. But for me, its characteristic to any large society that has an insular community living within them. You see it in the real world; its like how people started hating on Italians or Jews or Irish when they had their own neighborhoods.

Also, im 50% convinced one of the Dunmer in Windhelm is actually an imperial agent trying to stirr up trouble and inflame racial tensions to weaken Ulfric.

All of this just indicates to me that Ulfric.. is not a good administrator of his hold. Hes big on fighting, and ideals, but when it comes to the day to day boring administrative stuff he doesnt know what he's doing, because he spent his entire life either as a monk-in-training, a warrior of his warband (in service to the Empire or not) or in a Thalmor prison (which he was detained twice, for YEARS each time). He was only released the 2nd time because his father died and he was the heir of Windhelm.

(We can always have speculation as to why the heir of Windhelm was sent to the Greybeards, or why Ulfric's father never saw fit to appoint another heir).