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Both of them are often used interchangeably and in this case, the second one would sound more formal. Only the first one has some different usages/meanings.

仕方が無い:
① to have no way to do something, or not to find a good way to do something
② against your will, to do something that you can never change, to do something unwillingly/ reluctantly or to be unable to prevent yourself from doing something
③ used for someone who is lazy or causes an annoying problem
④ to be unable to put up with the urge to do something

やむを得ない: against your will, to do something that you can never change, to do something unwillingly/ reluctantly or to be unable to prevent yourself from doing something.ーIt has only these meanings.

仕方がない。その方法で試してみよう。←ok
やむを得ない。その方法で試してみよう。←ok

電車に間に合わなかったのは、仕方がなかった。←ok
電車に間に合わなかったのは、やむを得なかった。←ok

あいつは、いつも部屋を散らかす。仕方がないヤツだ。←ok
あいつは、いつも部屋を散らかす。やむを得ないヤツだ。←unnatural

友達に会いたくて仕方がない←ok
友達に会いたくてやむを得ない←unnatural