"I think it's trying to communicate...what do we do?" - Well-Known Meme.
I think he's simply asking if a Khajiit would join the Stormcloak Faction for the Civil War.
...And that he doesn't want it to turn into a racist discussion, about the fact that Ulfric hates elves etc.
To answer your question, though: Yeah, why not? It depends where the character's feelings truly lie; with the Empire, or the 'rightful' () Stormcloaks.
No one of any race should join the Stormcloaks. Ulfric is nothing but a narrow-minded idealist whose arrogance is only exceeded by his selfishness. He has no clue who the real enemy is, (the Thalmor) and as a result has to have a scapegoat. This scapegoat is the Empire.
I tend to think that this question can't be answered without a deeper glimpse into this Khajiit's backstory. Elswyr was taken by the Aldmeri Dominion, and a Khajiit may either support or dislike this. Certainly some Nords are racist (how often have I been called a carpet?), which is motivation to fight Ulfric, but many are good people. A mercenary would more likely join the Empire, based on the side more likely to eventually win as well as the side with more secure funding to pay auxiliary mercenaries, but the Stormcloaks' style of guerilla warfare is more suited to a Khajiit's skillset. And a criminal Khajiit may not want to join a side. Conflict breeds profits, and if guards are busy fighting each other they're not chasing you.