Which caffeine-free tea herbs have tannins?
Oak perhaps?

I know oak chips are added to wine and other alcohol for flavor (originally aging in oak barrels, but in this day and age oak chips are likely easier to source and use).  Oak adds tannins to the liquid, and the oak chips sourced for home-brewing are chosen to be food-safe, and the flavor is probably both generally good and also neutral, or perhaps I mean flexible, for it to work with so many different brewed drinks.  I see no particular reason a tea or extract can't be made of these oak chips, or such chips used to add tannic elements to other teas.

If you have access to oak trees, I think acorns might also work - bitter acorns are edible, but only after soaking until the tannins are down to a manageable quantity - the liquid is usually discarded, but that's because it is very strong, dilution might make a reasonable flavor.  The acorns are not poisonous, or else soaking would be about making them safe rather than palatable, and it takes a lot of soaking (several changes of water), so you should get quite a lot of... tannic extract? per volume of acorns that you can use to supplement other teas.