Streaming digit-to-digit conversion from decimal to hexadecimal
I don't think this is possible. Changing a high-order digit in a base-10 number -- say, changing 5000000 to 6000000 -- can change bits in its binary equivalent all the way from high-order bits to fairly-low-order bits:


10011000100101101000000
10110111000110110000000


It doesn't change the very lowest bits: Since $10 = 5\times 2$, the digit in position $k$ in a decimal number is a multiple of $10^{k-1} = (5\times 2)^{k-1} = 5^{k-1}2^{k-1}$, so changing it will never change the lowest $k-1$ bits of the binary representation. But it can change the $k$-th bit, and bits in positions up to $\log_2(9\times 10^{k-1}) = (\log_2 10)(k-1) + \log_2 9 \approx 3.322(k-1) + 3.170$. This is true for any $k$, so the range of affected digits grows indefinitely as $k$ increases.

If you were, say, trying to convert between base 4 and base 8, you could group each block of 3 base-4 digits into a base-64 "superdigit", which you could then convert back down to 2 base-8 digits since 8^2 = 64.

But there is no "right" superdigit base for the bases 10 and 2, since 10 contains 5 as a factor and 2 does not: every positive integer power of 10 contains 5 as a prime factor, and no positive integer power of 2 does, so by the uniqueness of prime factorisations there is no integer that is a power of both 2 and 10.