Yes: it gives the warforged a free tool integrated into its body, and expertise (double proficiency) with that tool
It's pretty clearly stated in the Envoy warforged's Integrated Tool feature description:
Choose one tool you’re proficient with. This tool is integrated into your body, and you double your proficiency bonus for any ability checks you make with it.
The feature on its own has you choose a single tool you're proficient with; the tool you choose is then integrated into your warforged character's body.
This is reinforced by the corresponding section of Keith Baker's FAQ on the 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.
Tinker’s tools weigh ten pounds! Can an envoy have a built in tinker’s tools? Or a full dragonchess set?
Certainly. The intention is that your warforged has a functioning version of this tool. This doesn’t mean that it in any way resembles the tools a human would use to perform the same task. A warforged with an integrated herbalism kit could literally have a garden built into its back, and pull out the herbs it needs when it needs them. Most important, consider that warforged are inherently magical creatures. They CAN physically alter their bodies (as shown by healing and integrated protection) and as shown by the spell prestidigitation it’s possible to create small, temporary objects through magic. So an integrated tool could be fully functional at all times—a warforged with smith’s tools has a hand that can function as a hammer. But it’s just as valid to say that the warforged does either perform a minor transformation to produce these tools or that it literally produces a temporary tool. Your integrated thieves tools could be lockpick fingers, but you could also study a lock and manufacture a key to fit it. Essentially, the FUNCTIONAL EFFECT is that the warforged always has a working version of this tool—but it’s up to you to decide exactly what that looks like.
The first quoted question mentions that the tool is granted by the feature and is part of its body; Keith Baker, creator of Eberron and author of the Wayfinder's Guide to Eberron, doesn't contradict this claim in his response. The answer to the second quoted question explicitly states that the integration grants the ability to use one specific tool at-will - whether that's flavored as the tool itself being built in to the warforged, or as the warforged having the ability to produce that specific tool (for personal use) at will.
In short, yes, it grants the warforged a free tool. If it didn't, it'd make no sense to only ask for proficiency in the tool as a prerequisite, rather than requiring you to have the tool itself.