I'm using EssentialsX to set the nickname of my player, but DiscordSRV shows the username on join instead of the display name. My guess is it is possible to use some Essentials API to get display name if it's installed?

Already fixed on v13

Have you set %displayname% here?
https://github.com/Scarsz/DiscordSRV/blob/master/src/main/resources/config.yml#L180

Yes, I am using %displayname%, haven't changed then from the default

Do you also have this set to true in EssentialsX's config.yml?
https://github.com/drtshock/Essentials/blob/2.x/Essentials/src/config.yml#L36

Yes, and it works for the leave command. Guessing it sets the display name after the join message is sent to discord?