Bitcoin wallet website
You'd have to get creative here.

Step 1) Get unspent outputs. https://bullpay.com/docs#GetUnspentOutputs
You can build your unspent outputs from multiple wallets. As long as you have the private keys, it can be done. 
Step 2) Build your signed transaction. https://medium.com/coinmonks/how-to-create-a-raw-bitcoin-transaction-step-by-step-239b888e87f2
Step 3) Send your signed transaction. https://bullpay.com/docs#SendTransaction or use a node.

The part you'll want to repeat is in step 2. Building out the inputs each using their own private keys. The tricky part here is waiting long enough to have multiple outputs to warrant going through all this.