Chances are the TIE pilot does have oxygen tanks at least. It's just not... Well, totally pressurized and whatnot, or something.
A suit like that can't carry enough oxygen for an operation. At all.
That's why Obi-Wan calls them short range fighters in ANH. Basically every source of canon (I know it's "legends" now) states that they have no oxygen inside, until the appearances in Rebels.