Lamport's one-time authentication scheme requires a one-way function. In terms of cryptographic hash functions, one-wayness is (practically, modulo some technical differences in the definitions) equivalent to first preimage resistance. In particular, a function does not need to be collision resistant in order to be one-way.
(A practical way to construct a non-collision-resistant one-way function is to take any given one-way function, such as a secure hash function, and make it ignore one of the bits of its input. This weakens the first preimage resistance of the function only slightly, but completely destroys both collision resistance and second preimage resistance.)
Indeed, the original description of the scheme (Lamport, 1981) does not mention hash functions at all, but rather suggests constructing a by encrypting a fixed plaintext with any secure encryption algorithm, using the input as the key and the resulting ciphertext as the output (citing Diffie & Hellman, 1976). Such a construction generally does not guarantee collision resistance and, in particular, will trivially lack it if the encryption scheme has equivalent keys.