Translation:Why does my boss not want to promote me?
You could switch their places and it would be just as fine, i.e."Varför vill inte min chef befordra mig?". The same goes for English as you could just as well say "Why doesn't my boss want to promote me?"
However, I would personally argue that the original version (in both languages) focuses more on the action (the lack of a promotion) rather than the subject (the boss). For example, "Varför vill min chef inte befordra mig?" seems to question why a promotion is not given to you whereas the other version seems more to question why your boss is not the one that gives you a promotion, as if you actually have gotten a promotion from someone other than your boss.