The general rule is to write words with auxiliary function in kana, so it's usually written
Searching the Balanced Corpus of Contemporary Written Japanese (BCCWJ, 少納言, http://www.kotonoha.gr.jp/shonagon), なくてはいけない has 902 results from books, newspapers, blogs, etc., whereas なくては行けない has only 4 results from internet sources (3 from Yahoo! ブログ, 1 from Yahoo! 知恵袋).
Other examples of words with auxiliary function would be とき (as in 小さいとき) or みる (as in やってみる).
なくては行けない only accounts for 0.4% of results, but 小さい時 for 41.9% and やって見る for 5.2%. I think that this reflects how literal the expression is still felt to be.
In any case, in your case it's quite clear that なくてはいけません should be written without kanji.