If I remember correctly, Luffy cut himself with a knife to show that he had the enough courage to join Shanks' party, right?
まで here is to indicate 極端な例 (see 1-4) of the act of 頼む. A full unfolding of the phrase would be 今日は顔にけがをしてまで頼んだのにシャンクスは航海に連れて行ってくれない.
(Edit: It may be that まで should be understood as indicating 動作の及ぶ範囲, but the explanation below remain more or less valid.)
A possible difficulty is that injuring oneself is not usually a way of asking. But in this scene, getting injured is actually to show the courage, so put it more verbose, 今日は顔に怪我をしてまで自分（＝Luffy）の勇気を示したのに・・・.
So the sentence translates like Even though I injured myself (to show the courage) and asked to take me to the sea today(, Shanks didn't agree to do so).
Hopefully the following examples help:
- 昨日は徹夜までして探したのに見つからなかった : Even though I searched without sleeping yesterday, I didn't find it
- 親に借金までしたのにうまくいかなかった : Even though I borrowed money from my parents, it didn't work out.
- 彼女と別れてまでアメリカに来たのに結局何も得られなかった : Even though I came to US leaving my girlfriend, I got nothing in the end.