Dev here, best way to do is use something like blockTrail on your website/application to generate addresses for your wallet with their API. If you have some cash and want a full explanation or sample codes or something integrated to your site lemme know(pm me) 
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. While relying on a third party for payment processing is probably okay in the majority of use cases, it's worth noting that a site that handles large amounts of money and for which availability is a concernlike the ones Neeo gave as an exampleshould implement it themselves rather than trusting a third party.
I agree with you on some level. The best thing about blocktrail is you can export your wallet from them (they can't access it directly since they lack your private keys) and then you can import your wallet elsewhere if you need to to use it again(if they went down for some odd reason). And yes if they dissappear you need to code your own RPC client code integration that can handle the wallet and post back and for with your site for new addresses/deposits and updates and so on.