Can a service that can only sign text, be used for encryption/decryption?
The signature algorithm would need to be deterministic. Only PKCS#1 v1.5 based RSA signatures are deterministic; most other modern (and popular) algorithms use random input for the signature. If the signature is random then your encryption and decryption key will be different and the protocol will fail.

The random blob of data isn't kept secret (if you could keep it secret between the two parties you would not need the signing service). Anybody can access the blob. So anybody with access to the service will be able to generate the signature used to derive the secret key.

So (after the edit) the scheme could be used to generate a symmetric key. But beware that whenever the private key changes or whenever the private key is used for a different, non-deterministic scheme it will fail. If the signature is deterministic you could however generate a static shared secret and use that for further communications.

It's kind of obvious, but remind your self that the owner of the private key service will also be able to generate the secret key. You might as well randomly generate a secret and put it somewhere in the service. Then you would not be dependent on the private key being available (using keys for different services is severely frowned upon when it comes to key management).