I don't know exactly about Reuters but often implied volas in the Black 76 world are quoted (forward) ATM. Thus the forward equals the strike and they dissappear from the formula:
$$
C = E[(F-K)^+] = \exp(- r t) (F N(d_1) - K N(d_2))
$$
and $d_1 = (\log(F/K)+\sigma^2/2T)/(\sigma \sqrt{T})$ and $d_2 = d_1 - \sigma \sqrt{T}$,
see e.g. here.
In the ATM case $F=K$ and in the term for $d_1$ we get $\log(F/K)=0$. Thus the formula depends as little as possible on curves. For $r$ I assume some appropriate money-market rate depending on the time-to-expiry of the caplet.
EDIT: I have worked using swaption data. There in the surface you have 2 dimensions: time to expiry of the option and then the term of the swap. Concerning the rate $F$ it is the traded swap rate that fits to the term (and the starting date) and thus is is a forward swap rate. The strike is then clear.
Summarizing: for the underlyings one should take the corresponding traded objects. In your case I would take a forward money market rate. If it is not traded then I would calculate it using the usual forward rate formula and take a money-market/swap based curve as basis (use zero-rates which you get by bootstrapping). Doing this you don't need a stochastic interest model for this but derive the underlying rather directly from traded objects. I am sure Reuters is doing something similar.