We don't know but he could've been.
The only information on the matter that I could find was the following extract from Rita Skeeter.
Now approaching his eighteenth birthday, Dumbledore left Hogwarts in a blaze of glory - Head Boy, Prefect, Winner of the Barnabus Finkley Prize for Exceptional Spell-Casting, British Youth Representative to the Wizengamot, Gold Medal-Winner for Ground-Breaking Contribution to the International Alchemical Conference in Cairo.
(Deathly Hallows, Chapter 19, The Life and Lies of Albus Dumbledore).
It looks as if Dumbledore was already on the Wizengamot when he left Hogwarts at age 17. Pottermore gives his date of birth as 1881, meaning that he was a Wizengamot member by 1896 at the latest.
There's no information in canon that I'm aware of that states whether Dumbledore was a permanent Wizengamot member from his time at Hogwarts until his ascension to Chief Warlock. On the one hand, the title of 'Youth Representative' suggests that the role was temporary, with one Hogwarts student perhaps being elected to the position each year to gain work experience. On the other hand, if anyone was put onto the Wizengamot that young then it would've been Dumbledore. If he did well during his time as Youth Representative then they may have offered him a permanent spot. This is mostly speculation, however.
So the data is inconclusive. We don't have any real clue as to what Dumbledore did between his sister's funeral and the events of Fantastic Beasts. Of course, it's likely that the films will give us more backstory on Dumbledore. All we really know is that he had become the Transfiguration teacher at Hogwarts before the events of The Crimes of Grindelwald, since he's shown at Hogwarts in the trailer. We know that he wasn't headmaster yet as Armando Dippet held the post during the opening of the Chamber of Secrets in 1942. There's no info that I'm aware of about who was Chief Warlock in the 1920s, or on Dumbledore's other positions and titles.