As of this date, No, a warforged cannot by RAW integrate an airship.
Due to the Player's Handbook chapter 5 (Equipment) sections 'Tools' (page 154) and 'Mounts and Vehicles' (page 155), it seems that vehicles are considered to be a type of tool, allowing them to be chosen for Integrated Tool. But they only state 'land or water' vehicles.
Further, while the airship is listed within the Dungeon Master's Guide (table on page 119), including being able to be owned by the players, it is not listed among the vehicles that the player can obtain proficiency in within the Player's Handbook.
Given that the airship is listed in the core 3 books, but is not among the options for proficiency, it seems players were not intended to have skill in its use.
As players could not obtain proficiency in air vehicles, they would not be eligible for the Envoy's Integrated Tool.
For this reason, I think the listing on DND Beyond is likely in error, especially as its listings for vehicle proficiencies are: 'Land', 'Sea/Air', 'Water'. The fact that there is a 'Sea/Air' option, as well as a 'Water' option, seems to indicate an error in the listings (since 'sea' and 'water' vehicles are the same).
Finally, while players could train (Player's Handbook, chapter 8 Adventuring, Between Adventures) to gain proficiency in airships, the text for warforged envoy does not indicate that the player can change what their Integrated Tool is, thus even after gaining proficiency in airships, they would not be eligible for Integrated Tool.