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Not using a card during a billing cycle has no effect on your report or score unless it's your only card. From the bank's standpoint, you'd have to go for several billing cycles before they start to care that the card has been going unused.
From a financial standpoint, it's a bad idea to charge something that you don't need. If you charge something at Wayfair for the purpose of keeping the card active, you're likely to be wasting money. On the other hand, it's easy to keep a major card active as you'd be buying something you would have bought anyway.
Is this the only card on your report?
If you have another card reporting a balance, then no, Wayfair reporting zero won't hurt our score. Having all revolving accounts reporting zero can cause a score ding so you should always have at least one card report a small balance.
I don't know about Wayfair's policy for inactive accounts -- but I highly doubt they'll close, or stop reporting on, an account for one month of inactive use -- maybe after 6 months to a year of inactivity.