Short answer
Long answer
PKI means "public key infrastructure", but with that public key comes the all important private key. The private key is the key used to sign (or generate) the certificates for your applications. Vault has nothing to do with it, it's the math behind the PKI that require it.
So the CA needs a key pair (public and private). Without the private key, it will be impossible to generate the signature and issue a certificate. It is computationally impossible to find the private key if you only have the public key (but ask again in 10 years from now).
Most PKI don't allow exporting their private key. Some/Most PKI are hooked to a tampmer proof hardware device (called an HSM) designed to prevent the private key to ever leave its secure container.
So unless you can get your hands on the private key, you will have to deploy a new issuer certificate. You can have Vault's certificate signed by your old CA. Doing so will make your certificates recognized by browsers, but now you have 2 CA to maintain...