For stream ciphers, IND-CCA1 and IND-CPA security differ precisely in that an attacker can choose the IV in the CCA1 game (because that's part of the ciphertext that can be submitted to the decryption oracle); but in the CPA game is constrained to whatever choice of IV the cryptosystem makes.
We can artificially construct a stream cipher vulnerable under CCA1, but not under CPA: consider AES-128 in CTR mode with random IV, except that the AES encryption block is modified so that for all-zero input block, the output block is the key. Decryption no longer works for the block cipher, but it still works for the stream cipher (albeit differently when the counter is initialized to all-zero, or reaches that). Under the CCA1 game, an adversary recovers the key by submitting to the decryption oracle a ciphertext with all-zero IV and first block of ciphertext; while there's no way to take advantage of the modified block cipher under the CPA game, and the stream cipher remains secure (the adversary has negligible odds to get the IV to reach zero, thus the stream cipher is indistinguishable from true CTR mode, which is IND-CCA1).
For a less artificial example, we can hypothesize a block cipher vulnerable to key recovery under CPA, but not under random Known Plaintext Attack. That weakness directly makes the derived CTR or OFB stream cipher vulnerable under CCA1, but not under CPA.
Thus if there was a block cipher vulnerable to key recovery under CPA but not random KPA, with the corresponding CTR or OFB stream cipher common, that would be a common stream cipher not IND-CCA1, answering the question in the negative.
However I know no common stream cipher that's vulnerable under CCA1. Much to the contrary, modern native stream ciphers like ChaCha20 and Salsa20/20 aspire and appear to be IND-CCA1. And modern block ciphers are believed secure under CPA, and their corresponding CTR stream cipher is accordingly believed IND-CCA1 (just as claimed in that other answer, even if I fail to follow it's proof sketch).