I can't think of anything in Vauxhall specifically, but in the area:
- River Effra - an underground river (and CSO?) that runs under Vauxhall, not a tunnel as per se though.
- Pimlico-Battersea steam tunnels - tunnels that go from the other side of the river at Pimlico, over to Battersea. Not sure how these have been affected by the Battersea Power Station re-development though.
- Clapham South Deep Level Shelter - tunnels beneath Clapham used as WW2 shelter, I believe they were then purchased by Iron Mountain for use as a document storage facility.
Sometimes it's worth looking at old Ordnance Survey maps, the one I've quickly looked at doesn't have a great deal of interesting leads - but I would have a guess that originally both the Bethlem and Workhouse would've had service tunnels. These are more towards Elephant & Castle - and the Imperial War Museum and Cinema Museum(?!).
As for Vauxhall itself, depending upon how long ago your contact visited the tunnels, I'd bear in mind that the development of Vauxhall Cross in to the SIS building may have likely changed anything that ran too close. (i.e. The Effra outfall runs directly under VX) Not sure where the new American embassy is, but Vauxhall rings a bell - and I would guess there would be similar concerns there.
Anecdotally, I seem to recall being pretty drunk and ending up at a bizarre place under the arches - opposite Vauxhall Cross - a few years ago as it was the only place open. I got lost looking for the toilets and remember poking around a basement area, which seemed to have an interesting doorway... or that's just a drunken mis-memory! So it wouldn't be surprising if anything underground was also used as storage - i.e. Mark Lane tube station at Tower Hill is partially used as storage for an All Bar One above it!