Currently the most popular web wallet that can be used on website for sending/receiving payments using their API is blockchain
But from last few months many blockchain api users are facing problems like missing payments, late confirmation of transaction etc.
Try to use block.io api. Most of the programmers prefer to use this these days and they are stable too
Another option can be bitpay where they charges 1% fee but it is the best
The reason for the "missing" payments is that they forward the payment without a fee. You can imagine that this costs them a lot and that's why they combine these forwards into one transaction. Hence it can take a while because they wait for multiple transactions before they send the bitcoins to your address. Sometimes I had to wait a few days in the past but the payment was never really lost.
They invented v2 of their API where payments are not forwarded anymore, instead they now use xPub keys to generate addresses which are linked directly to your wallet. Imho this is way too complicated. Also you must apply for their API now and they may reject your application. v1 is not even supported anymore, though it still works. They might just drop it soon, so I really suggest to not use it anymore.
The best is to make your own API but I can tell you that it's a lot of work if you lack of programming experiences.