HW acceleration for Camellia cipher (x86_64/AES-NI/AVX2)
The obvious answer to your question is "yes". The kernel mode implementation pointed to by otus clearly shows that it can be done.

That it can be done doesn't mean it gets done however. Many Google searches for sourcecode don't show any OpenSSL code that implements this functionality. In general, OpenSSL doesn't rely on the crypto code of the kernel. So, that means the hardware acceleration is not available. In general you should not expect that hardware-crypto availability is automatically reflected in (higher level) API's.

To resolve this you could either implement an engine yourself or ask for a feature request at OpenSSL. Note that Camellia interest seems to be dwindling. Nobody picked up the - relatively new - feature request for Camellia in authenticated ciphers or ciphersuites so they got closed.

It seems that Camellia is seen as a "vanity algorithm" by many; let's not do AES because it is too American, but do something almost identical instead.