I'm not sure how to set up accounts (names, info) to match the client invoices with payments made to the ICs. Any advice would be appreciated.
To answer your question, you would need to turn on billable expenses in company settings, and if you want set a mark up.
Then when you pay the IC, select the customer that bill applies to, and when you create an invoice for the customer you can click add to bring that expense with or without markup to the invoice.
You make payments to the IC's as an expense regardless of whether or not the customer pays the invoice, and you charge customers with an invoice, that is your income,