The answer is: one, mostly. That is, most Reform congregations in North America celebrate Rosh HaShanah for one day. On the other hand, some North American Reform congregations, along with Progressive Jews in Israel and elsewhere, follow the traditional two-day observance of the New Year.
Why? Let’s begin with Leviticus, chapter 23, where we find a detailed calendar of the Jewish festival year. The text instructs us to “declare” each festival on a specific day of a specific month (verse 4). For example, the festival that we today call Rosh HaShanah occurs on the first day of the seventh month, which we now call Tishrei (vss. 23-25). In ancient times, the beginning of each month was determined through eyewitness testimony (Mishnah, Rosh HaShanah, chapters 1-2). Witnesses would testify before the beit din (the Rabbinic Court in Jerusalem) that they had seen the new moon. The Court would announce that a new month had begun. Unfortunately, communications being much slower back then, news of the Court’s announcement could not reach distant communities prior to the onset of that month’s festivals – for example, Sukkot, which begins on the fifteenth day of the seventh month (Leviticus 23:34). If these communities did not know the date of Rosh Chodesh, the beginning of the month, how could they count the fifteen days to proper date of the festival?
The solution was a practical one. Since the new moon can appear either 29 or 30 days after the beginning of the previous month, the distant communities began to count fifteen days from both of the days on which Rosh Chodesh might occur and to observe both of those days as the festival day of Sukkot out of “doubt” (misafek) as to which day was really the proper date. They did the same with other festivals in other months. This custom (yom tov sheni, the “second festival day”) has remained in effect in traditional Diaspora Jewish communities. In the land of Israel, by contrast, where news of the Court’s new moon announcement could reach the community prior to the onset of the festivals, the festival days remained a one-day observance.
Rosh HaShanah, for reasons too detailed to go into here, is an exception to the above. It is nowadays observed for two days in the land of Israel, as well as in the Diaspora.
The Reform Movement, from its beginnings in the 19th century, reinstated the Biblical standard of observing each festival day, including Rosh HaShanah, as one day. Since we no longer rely upon the eyewitness system for determining the calendar, the reasons given for maintaining the “second festival day” (Talmud, Beitzah 4b) are no longer compelling for us. Yet as mentioned above, some Reform congregations in North America have chosen to observe Rosh HaShanah for two days for a number of reasons, such as the desire to recover the traditional observance or to be in solidarity with our fellow Jews in the land of Israel and all over the world. It’s a very good example of the pluralism and the variety of Jewish life in the Reform Jewish community.