in response to the earlier thread (now closed), i was recently watching How it's Made and, twice, the process of hot dyeing was mentioned; thoughts a-rekindled, i've just had google and there are indeed hot processes and products available, though they tend to be more industrially biased.
further googling along these lines revealed people who've dyed vinyls with normal dyes, but by putting their items through a heating process. about 100C seems to be the key, though some seem to have done it at lower temperatures (60-ish).
also, and possibly cheaper overall (considering a need for trial and error), there are aerosols available:
how these dyes will affect the translucent and flexible materials typically used for 'baby pants', i don't know, but one source mentions a brittling of the material due to heat. also, and as i previously thought, the aerosols use acetone as an agent (to slightly molten the material surface in order to allow pigment penetration, no doubt).
anyway, here's me hoping to try the aerosol as soon as i can afford it. good luck to anyone else who tries it before me.