One of the thirty-nine categories of forbidden activity on Shabbat is writing. Rabbi Yishak of Vienna (late 12th-early 13th century), in his work Or Zarua, claimed that the Torah prohibition of writing on Shabbat applies only to writing “Ketab Ashurit,” the Hebrew letters as they are formed in a Torah scroll. In his view, writing in other languages, or in Hebrew but in standard form, as opposed to the form used in the Torah scroll, is forbidden on Shabbat only by force of Rabbinic enactment. A number of other authorities held this view, as well.
This ruling is of great importance with regard to the question of asking a non-Jew to write on one’s behalf on Shabbat. A famous Halachic principle permits asking a gentile to perform forbidden activity on one’s behalf on Shabbat in situations of “Shebut De’shbut Be’makom Hefsed.” This means that the forbidden act in question is prohibited only by force of Rabbinic enactment, as opposed to Torah law, and that the individual will incur a financial loss if that act is not performed. In light of the Or Zarua’s position, Hacham Ovadia Yosef ruled, in his work Yabia Omer, that one may ask a gentile to write something on Shabbat if this is necessary to prevent a financial loss. Hacham Ovadia maintained that we may rely on the Or Zarua’s classification of standard writing as a Rabbinic prohibition to permit asking a gentile to write something to avoid incurring a loss. One possible example of such a case is where a person must urgently receive a certain delivery and has to fill in a form. If he will incur a financial loss by not filling in the form, he may ask a gentile to fill in the form on his behalf on Shabbat.
Summary: Although generally one may not ask a gentile to write or perform other forbidden activity on Shabbat, one may ask a gentile to write something if this is necessary to avoid a financial loss.
See Menuhat Ahava, Helek 3, Perek 22, Halacha 8.