The Shulhan Aruch (O.H. 306:11), based on the Gemara (Bava Kama 80a, Gittin 8a), rules that one may ask a non-Jew to sign a document on one’s behalf on Shabbat for the purpose of buying a house in Israel. The Rama (ibid.) adds in the name of the Or Zarua, that if the document is signed in a language other than Hebrew, then the prohibition is only Rabbinic, and therefore, for the Mitzvah of buying property in Israel, it is permitted to ask the non-Jew to sign on one’s behalf. It should be noted that the Gemara does not draw a distinction between Hebrew and non-Hebrew writing and implies that both are prohibited biblically. Nevertheless, the Or Zarua does learn it this way and there are scenarios in which his approach is used to rule leniently.
For example, the Sephardic Aharonim discuss whether one can rely on the Or Zarua’s approach in the case of significant financial loss. Rabbi Aharon Alfandri (Yad Aharon, 306) writes that one may have a non-Jew sign documents on one’s behalf in a foreign script in order to prevent a financial loss. Similarly, the Peri Megadim (§ 444) writes that if one forgot to sell one’s Hametz to a non-Jew on Erev Pesah that falls on Shabbat, one could ask a non-Jew to sign the proper sale documents to avoid forfeiting all the Hametz.
Furthermore, the Ben Ish Hai (Rav Berachot, Ma’arechet Shabbat, pg. 145) discusses having a telegram that is sent on Shabbat signed on one’s behalf to avoid a financial loss. This can be applied to the modern-day scenario of having a package or document sent to one’s home on Shabbat and telling the delivery person to sign for the package on one’s behalf in a case where not doing so would involve a loss. There is further room to be lenient if the worker signs for the delivery digitally. Additionally, signing nowadays is often more of a scribble than it is distinct letters which also may not be considered Halachic writing.
Rabbi Ovadia Yosef (Yabia Omer, vol. III, Orah Haim, § 23) writes that that when he served as rabbi in Egypt, the practice was for a non-Jew to sit in the sanctuary and write, ostensibly in Arabic, the pledges of the congregants who donated money on Shabbat. Although he says it is not ideal, he relies on the aforementioned opinion of the Or Zarua to permit such a practice when necessary.
Summary: In a case of great potential financial loss, one may ask a non-Jew to sign a document on one’s behalf on Shabbat.