The vapor barrier/door seal is a large, form fitting piece of flexible foam plastic that is sealed to the door frame by butyl tape. The stuff I have is made by 3M (part no. 051135 08612) and is a 3/8", 15' roll. You may want to buy the 1/4" diameter.
This black goo (we used to call it dum dum) is what seals the vapors barrier to the door frame. It is needed because water gets into the door and drains out the holes in the bottom of the door by design.
NZ BMW-If water is appearing inside the rubber door seals, the most likely cause is a break in the vapor barrier seal. Of course, there could be damage to the rubber door seals but you would see that right away. Even though you don't see it, water goes through your front doors too and drains out the bottom.
Topaz-Water should not be dripping out from behind the door cards. Looks like you're right-the tint guys messed it up. Reseal your vapor barrier before that water destroys your door cards.