Can ChaCha be turned into a collision resistant hash function by xoring keystreams?
The only expectation for each $c_i[m_i]$ is that it's independent and uniformly random. ChaCha20, with the nonce/counter words acting as table index is perfectly fine here, under standard assumptions about ChaCha20. Furthermore, the $m_i$ are expected to be small—in RFSB, they are 8-bit wide—so you can put the message in the counter just as well.

As an example of using a larger message as input to each function, you may also look at Rumba20, $$\text{Rumba20}(x_1, x_2, x_3, x_4) = f_1(x_1) \oplus f_2(x_2) \oplus f_3(x_3) \oplus f_4(x_4)\,,$$ which instead uses different constants to differentiate between the 4 different input blocks, and uses 384 bits of each Salsa20 block as input. 

Note that if you let the number of xors ($w$ in the RFSB paper) grow too much relatively to the output size ($r$), the compression function can become weak. This is discussed at length in Section 4.

Note also that either of these compression functions only attempt to be collision-resistant. Other properties, such as preimage resistance, need to be ensured by some other mechanism, as the papers stress.