I have a database of OpenFlights data laying around from What's the quickest route between antipodes using regularly scheduled transit?, so let's put it to work.
[an excruciatingly boring period ensues wherein we discover that PostgreSQL got upgraded on my system, leaving a database that is incompatible with the new version, and entirely too much nonsense is involved in sorting it out.]
After preparing a scratch table:
SELECT ST_X(source_geom) as src_x, ST_X(destination_geom) as dst_x,
abs(ST_X(source_geom)-ST_X(destination_geom)) as delta, * from flights
order by delta limit 20;
For some reason, OpenFlights thinks that "Illinois Airways" operates a flight that both arrives and departs PKN airport in Indonesia. We'll remove that, along with duplicate/reverse city pairs, including those operated by multiple airlines/codeshares to get a few top results:
This gives our winner, flights between Baie-Comeau Airport and Mont-Joli Airport, both in Quebec, Canada. FlightAware tells me this is Air Canada Jazz JZA8968/JZA8964, nearly a straight shot north-south across the St. Lawrence River, a distance of 59km and a difference of just 0.0037 in longitude.
This is entirely based on the coordinates of the airports in the OpenFlights database, not the actual route of the flights, which will vary due to the usual flight planning considerations.
Update: I was curious what happens if instead of directly comparing coordinates, we go by azimuth instead, as JonathanReez suggested. This method gives us CGK-SGN (Jakarta-Saigon/Ho Chi Minh), the runner-up from the coordinate-based approach, with a bearing between the airports of 179.986.
This approach also produces candidates with greater degree differences closer to the poles, such as YQR-DEN. The bearing is -179.970 even though the coordinate difference is about 25", since the lines get closer together as you approach the poles.