Ask your DM
There seems to be a split between the RAW and the Rules as Intended. Ask your DM if you want to do this and see what they want to go with.
The Integrated Tool feature says:
Choose a tool you're proficient with. It becomes integrated into your
body, and you double your proficiency bonus for any ability checks you
make with it. You must have your hands free to use this tool.
So, to qualify for Integrated Tool something must be 1) a tool that 2) you are proficient with.
Vehicles appear to be listed as tools. If you look in the Sailor background for example it says:
Tool Proficiencies: Navigator's tools, vehicles (water)
And the last entry in the Tools table in the PHB (p 154) says:
Vehicles (land or water)
The slight ambiguity comes from the fact that Vehicles are then described in their own section apart from all the other tools, an indication that maybe the rules treat them as different from normal tools. However, it seems that when you take everything together that this objection doesn't really negate the fact that vehicles are literally listed as tools in the table and are listed as tools in proficiencies as well. So, the rules seem to consider them tools.
So, if your character takes a background that gives you a vehicle tool proficiency (such as sailor or soldier), then you should meet both requirements and technically should be able to use it for your Integrated Tool feature.
Design intent: No (ask your DM)
According to Keith Baker's FAQ on Wayfinder's Guide to Eberron:
An envoy warforged has an integrated tool that’s part of its body, and
gains expertise with that tool. In the PHB, tool proficiencies
provided by backgrounds include vehicles. So can I have a warforged
with a built-in wagon?
All things require the approval of the DM. If your DM WANTS to let you
have a warforged with a built in boat, that’s great. But the design
intent is that this applies to the specifically identified tools
listed on page 154 of the Player’s Handbook. This does not include
mounts and vehicles, which are detailed on the following page
(“vehicles” are on 154, but not detailed). So again, if you and your
DM agree, go for it. But it is not the design intent.