Accepting Bitcoin payments without the full blockchain or third party APIs?
If you don't want to run a node or use any 3rd party APIs you can use a library since you only want to accept payments.  

Use an extended public key from one of your wallets, and generate new addresses from the HD xpub using a library. When a user pays it will show up in your Phone/PC wallet. 

For example with Bitcore library it would be:

var bitcore = require("bitcore-lib");

var pub = new bitcore.HDPublicKey('xpubMyHDPublicKey');
var derivedHdPublicKey = pub.derive('m/0/1'); 
//increment up the keychain m/0/2, m/0/3, etc each order
var derivedPublicKey = derivedHdPublicKey.publicKey;
var address = new bitcore.Address(derivedPublicKey);
console.log(address);


Then just keep track of orders and addresses on your server database.