Answer: There are a few potential issues with making sushi on Shabbos.
The Gemara (Shabbos 145b) teaches that one mustn’t rinse salted fish on Shabbos. The Levush (OC 318:4) and Pri Megadim (OC 318:16) explain that doing so would be makeh bepatish (See Aruch Hashulchan OC 318:29). The Mishna Berura (Biur Halacha 318) disagrees, and writes that makeh bepatish does not apply to food (See Igros Moshe OC 3:52). Regardless, R’ Asher Weiss (tvunah.org) writes that as each of the ingredients are edible before rolling, this prohibition wouldn’t apply even according to the Levush and Pri Megadim.
The Chayei Adam (87:2) writes that the prohibition of tefira, attaching, also applies to food. Nonetheless, moistening the seaweed sheet to close the roll would not pose an issue of tofer as it isn’t something that lasts (See Rema OC 317:3). Additionally, R’ Weiss notes that a little moisture does not count as attaching.
The Shulchan Aruch (OC 321:12) writes that one must not cut vegetables into small pieces as doing so is tochen, grinding. Thus, the Mishna Berura (321:45) writes that one must ensure that the pieces are slightly larger than usual. Nonetheless, he cautions (Biur Halacha) that this is hard to define.
The Magen Avraham (340:17) writes that one mustn’t create cheese on Shabbos, as it is included in the prohibition of boneh, assembling. While some foods can be attached together, one mustn’t do so to create a particular shape or picture. Following this, the Kitzur Shulchan Aruch (80:25) and Chayei Adam (39:1) write that boneh applies to food as well as other items. Based on this, R’ Asher Weiss writes that creating sushi on Shabbos would also be an issur of boneh.
In conclusion, one should not make sushi on Shabbos.