What type of bird has the greatest wing area?

Birds with largest wingspan

Wingspan Length

Wandering Albatross 8.2 to 11.5 feet
Marabou Stork 7.4 to 10.5 feet
Andean Condor 8.9 to 10.5 feet
Pelican 7.9 to 10 feet
California Condor Up to 9.8 feet
Jabiru Stork 7.5 to 9.2 feet
Kori Bustard 7.5 to 9.02 feet
Whooper Swan 6.75 to 9 feet
Great Bustard 6.9 to 8.9 feet
Trumpeter swan 6.7 to 8 feet
Mute Swan 6.5 to 7.8 feet
Black Vulture 4.5 to 5.5 feet

http://birdsflight.com/bird-largest-wingspan-world/

The wingspan is the distance between the two wingtips. So the largest wingspan does not necessarily correspond with the largest wing area, which is also determined by the shape of the wings. With that said, the Magnificent Frigatebird (Fregata magnificens) is said to have the greatest wing area.

https://en.wikipedia.org/wiki/Magnificent_frigatebird

Here is the wing area for the Wandering Albatross...

Wing Area = Wing Length * Wing Width

Solving for Wing area:
340 cm = Length of a Wandering Albatross Wing
15.7 cm = Wing Width for Wandering Albatross
340 * 15.7 = 5,336.5 cm²
Wing area = 5,336.5 cm² or 827.159154 in² or 5.744160792 ft²

http://0310fcb.netsolhost.com/curric/teachers/t_profiles.htm

Wing Area = Wing Length * Wing Width

This formula suggests that the bird's wing is a rectangle. I would venture to doubt it.

https://aerocorner.com/wp-content/uploads/2019/06/bird-and-aircraft-wing-shapes-wizes1.jpg