From what I see you have 2 options:
1. Use anonymous altcoins, but then your service may reach smaller number of users.
2. If you want to use Bitcoin, then you may need to use a (reputable) mixer to fund your sending addresses.
However, whatever you do, I advise you use proper local wallet, where you have access to your private keys, seed, or whatever you need to clone that wallet if you need that (like hardware crash).