Is there a website anywhere where you can find cheap last-minute train tickets?
Almost any website will sell on the day Advances if they are available. But you may not find them "last minute":-
. Generally it's up to an hour or so before departure from the origin station.
Do VTEC or others sell off tickets cheap to try and fill the train?
Not really, no.
XC do have some fairly cheap on the day Advances for some of their shorter distance flows (e.g. Stafford to Birmingham) at very short notice, this tends to be more likely to occur on flows where they are in 'competition' with other train companies (West Midlands Trains in that example).
My parents had thought about going down to London for a day from Leeds but the cheapest they could find at this stage was £116.
Trainsplit is offering a £78.51 return fare tomorrow, arrriving into KGX 1114 and departing 2035 (in value mode), or if just an afternoon in London will do, then it can be done for £58 return.
It is better to book further in advance (with the caveat that the 20th May timetable change is causing problems on numerous routes)