Oracle Wallet Manager and orapki do not let you extract the private key associated with user certificate located in Oracle Wallet. If you need it for some reason, for example testing with external tools like wget, then its possible to extract the private key using openssl, since Orale Wallet (ewallet.p12 file) is just PKCS#12 file.
[oracle@jfadboc1n01 wallet]$ openssl pkcs12 -in /home/oracle/wallet/ewallet.p12 -nocerts -out private_key.pem Enter Import Password: MAC verified OK Warning unsupported bag type: secretBag Enter PEM pass phrase: Verifying - Enter PEM pass phrase: [oracle@jfadboc1n01 wallet]$ ls -l total 16 -rw-r--r-- 1 oracle oinstall 11629 May 15 11:38 ewallet.p12 -rw-r--r-- 1 oracle oinstall 1879 May 17 08:53 private_key.pem
Here private_key.pem contains the private key extracted from Oracle Wallet.
The first password that openssl asks (Enter Import Password) is the wallet password, the other password (Enter PEM pass phrase) is used to protect the exported key.