As a student with questions about a paper, can I email the author myself or should I ask someone else to contact him? I am reading a paper and have questions about the details of the procedure described. I have read other papers by the same team but they don't explain too much about that procedure anyway. I think it might be common, but my supervisor doesn't know it too.

I am stuck and I want to get out of it. As a student working on my master thesis, can I email the contact author for the manuscript cold out, or should I ask someone to contact him for me? I would ask my supervisor but I don't want it to imply that I'm avoid taking initiatives when I could do it on my own.
You can definitely contact a paper author. They might be of the 'it obviously follows' == after 5 pages of calculations kind, or the empiricist who published the 20 successful regressions or simulations out of 200, with 180 contradicting their result or being inconclusive; and in either case ignore your question. From personal experience though, it can even lead to breakthroughs: in my case, someone sent me his lecture notes which clarified something I was stuck with, and related to the submitted question. However, if your advisor knows the author, or simply is well-known in their field, do mention that you are their student, as it should increase good will on the author's part - after confirming with your advisor that they are cool with it. Showing that you are active, interested, and independent should also go down well with the advisor.