You are looking for an encryption algorithm which produces ciphertexts that are completely indistinguishable from random. Your comments:
I've been looking at the "off-the-shelf" recommended encryption algorithms and they all seem to have associated data.
Actually, that's not a problem. Yes, many encryption algorithms support associated data; however that's meant to be a facility for you to use; there's no requirement that you use it. You can have empty associated data without any issue.
You also bring up the authentication tag (which is data that is used to verify that the ciphertext hasn't been modified in transit) - since that is typically indistinguishable from a random string, it does not conflict with your requirements.
Now, one thing that may show up as nonrandom is the nonce (also known as IV) - some modes insist that we never repeat the nonce, and so what is common practice is to use a counter to generate successive nonces when we generate ciphertexts - as such, such a successive counter may show up as nonrandom.
The way to address that is easy: use a mode that doesn't require such nonrepeating nonces. Such modes would include CBC mode (with random IV's; you might want to include something like an HMAC or KMAC to the ciphertext if you care about detecting changes to the ciphertext) or an SIV mode (which generates its own IV, and in a manner which appears to be random). OpenSSL (actually, any reasonably complete crypto library) would do CBC mode for you (and possibly SIV as well).
BTW: the terminology of what you're trying to do (hide a message within an innocuous appearing larger message) is called Steganography; you might want to google that for ideas on how to embed this random ciphertext into your HTTP messages (images, html text, whatever)
