Do followers of Avraham Weiss’s Open Orthodoxy have their own siddur or use someone else’s? I’d be curious about any ways their liturgy might differ from the mainstream.
Many of the people who daven at the Hillel Minyan at Northwestern University (where I davened in college) associate with the Open Orthodox movement. Most of them daven with the Koren Sacks siddur. Some others use the Artscroll Siddur edition that includes the prayers for Israel, TzaHa"L, the United States, etc.
Since they use mainstream siddurim, their liturgy does not really differ from the mainstream. Most of the difference between them and more mainstream Orthodox Jews is difference in interpretation in hashkafa/halacha (depending on who you ask).
Although there is not yet a dedicated Open "orthodox" siddur, those associated with this label tend to have a problem with the beracha of "shelo asani ishah." This was the topic of volume 2 of the Yeshivat Maharat journal, Keren, and it has also been discussed repeatedly on Morethodoxy, which is apparently another name for the movement. See, e.g., here, here, here, and here.