Is there a way to send bitcoins programmatically without the bitcoin client using PHP?
You will need to connect to a bitcoin node in one way or another to broardcast to the newtork. There are a number of php and bitcoin open source repos that will do this, have a look here https://github.com/search?q=bitcoind+php&ref=cmdform
Another option if you do not want to run the bitcoin client locally is to connect to an api to broadcast your transaction.
An example I found for php connecting to blockchain.info is as follows
message; $txid = $json_feed->tx_hash; ?>
Hope this helps
UPDATE: To run requests to the blockchain.info API you need to follow the setup instructions here https://www.blockchain.com/api/blockchain_wallet_api
NOTE To use this API, you will need to run small local service which be responsible for managing your Blockchain Wallet. Your application interacts with this service locally via HTTP API calls. Click here for complete setup instructions on GitHub.