I just went and looked at it and crawled under the van.
The socket thing is not needed. Just the long black 12mm Allen wrench. It goes about 2/3 maybe 3/4 into the hole under the black cap. There is a black cap to remove, above the bumper, kind hidden from view beneath the rubber seal, to the right of the center of the bumper.
Looking under the body. There is a black thing that the hex thing goes into ... it is an odd shaped box, that holds a reel of steel wire and then has a sheathed cable guide coming out one side, so the thing is like a manual cable winch. The back side of it, with a metal cover, facing the tire doesn't have a backup knob or tool access, no holes. There's two hex nuts, on the cover side, but also a bunch of metal shaft ends ... I don't know if that cover facing the tire would come off, if one were to undo those two nuts ... certainly could be dangerous to try to take that apart and the whole thing might come undone, like a seat belt retaining assembly?
It may be feasible to remove the whole winch assembly. It is held in place with three nuts, they're not too big, but they have the shape of the end of a Torx screwdriver. After removal, you could then engage it probably with a stubby 12mm Allen wrench. Probably safe. I have not seen that kind of nut - nor a nut socket to remove it yet. (not a mechanic, just a driver)
Removal of the tire plug that holds it up seems a total no go. There's a small metal element with spring that holds up the center of the wheel. No nuts, parts. anything that could be tampered with.
Options:
- borrow the long part from a dealer? call dealer?
- find a driver of the same vehicle to borrow it? ( are there any other cars with this setup ? )
- do parts stores that loan out tools carry something like this extra long 12mm Allen wrench?
Alternatively borrow a similar size tire, exchange the tire, drive to dealer to borrow the tool
As it looks rather specialized, I was looking online and did not have much luck finding something that long hex 12mm (you can buy bare steel hex rod that size )
Also, depending on the issue, hole / tear / ..., a can of fix-a-flat, drive off to the tire shop to have the tire patched.
Then you don't need the special tool yet.
Official tire removal is shown in this video ...
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