Can the office that orders lunch every week charge it to their house account on my ordering page?

Not today. A house account is spent at the counter: your cashier picks House account on the Charge screen and chooses the company from the list. On your ordering page the only balance a customer can use is a gift card code, and everything else is cash or the card machine when the order arrives. So the office rings you or orders at the counter and the sale goes on the account, and you record their payment later under Settings → House accounts. The Gift cards, store credit and house accounts tab in the docs describes both sides.