Is there a thing called "Open Source Web Wallets" ? Welp! I need to look that up son!
Would really help figure out transparency behind the wallet.Most of the online wallets do provide built in API's but with limited access and cross verification of the tokens.I'd rather write my own API instead of begging them to provide me unlimited requests per minute.
Check coinb.in they are opensource web based bitcoin wallet but i don't think they have API option.
You have to run own bitcoin node (wallet) to create own api or you can simply call your node using jsonrpc.