You should store and keep the private key with you. The private key gives you access to move the funds of a given address. There is no way to get the private key using this API.
If you want to send a transaction using blockcypher, they have an endpoint for that, but you'll need to sign the transaction before using it.
The following tutorial might help you: https://blockcypher.github.io/documentation/samples/create-tx.html