The only difference between Altmer and Chimer were the gods they worshipped, correct? So would an Altmer who revered the Daedra and ALMSIVI be allowed to join the Tribunal Temple?

Worshippers are currency to gods. And all gods want as much currency as possible.

I don't care about the gods themselves. It's not like when you ask to join the Temple Vivec comes out and says you're alright to join. Would the priests raise an eyebrow at an Altmer wanting to join the Temple?

I'm certain there are probably Altmer who are lay servants to the Temple, although perhaps not so much after the events of Oblivion, which weakened the Temple's influence even on Morrowind.

Well, this is for a character to be played in Morrwind, so that's all good. I wanted to make a holy battlemage character, that would join the Temple and House Redoran, and being a Dunmer was naturally the first choice, but I really want an excuse to play a race with more magic. Heh.

The priests do whatever is in their respective gods best intent or follow the respective gods commands/tenets. So the question you should ask yourself is, what would the god think?

It was never said anywhere in Morrowind that race had any bearing on whether you could join the temple or not, as far as I know, and you could join the temple with any race in the game. While this may of course be for gameplay reasons, unless it is stated otherwise I think we can generally assume what we see in the game fits the lore. Besides, there is no logical reason to deny an Altmer entry to the temple if said Altmer is truly devoted to the temple's belief, not that logic means much when talking about Dunmer or the Tribunal, not the same kind of logic anyone else follows, at least, but considering all the factors I have noted, it seems unlikely that the temple would not allow an Altmer to join.

Aren't there two Altmer temple scouts in Molag Mar?

Mercenaries.

Even so, I'd say the the glorious thievery of Tribunal is pretty similar to what many Altmer desire anyway, that is, ascendance.