Non- Nords have a hard time justifying their joining the Stormcloaks, at least in my opinion. Anyone can join the legion and easily fit in.
As I see it:
Nords - can join either side easily.
Imperials - Stormcloaks possibly by having some Nord ancestors, religious convictions or something like that.
Bretons - Could go Stormcloak with the Nord ancestors, religious convictions or pro-Forsworn that wants chaos to allow the Reach to break free
Redguard - Could feel sore about the empire and join the stormcloaks to get back, or the ancestors/religious bit.
Argonian - could go stormcloak to foster better Nord/Argonian relations, might just dislike the empire.
Khajiit - Could be an undercover Thalmor agent trying to weaken the empire. Maybe some sort of rabid empire hater. hard to see another reason considering how racist Nords are and how Khajiit are not allowed in cities.
High Elf - I'd find it hard to believe that the Stormcloaks would let an elf in in the first place. Getting past that plot hole you could go as an undercover Thalmor agent trying to weaken the empire.
Dunmer - Could join as a Riften style dark elf that has integrated with ord society as opposed to the Windhelm ghetto dunmer.
Orc - Could join as a supporter for more independence for the Orc strongholds.
All in all you need to comeup with more of a backstory to support the Stormcloaks as a non-Nord, joining as a non-human is even harder. Too bad one can not join or at least support the Thalmor in game, that would have been a nice option especially if they had incorporated a faction system more similar to fallout new vegas.