The floor covers I've seen have been brass with the round covers for the receptacles. The covers twist/screw on when not in use to hide the receptacle. The link below shows a few different varieties.

In the link they have one with a flip up cover. Might be good under a table.

Here is a link that might be useful: in floor covers

Thank Cissado. The link is exactly what I want but the price surprises me. Checking a few other links suggests none are in the range I hoped to pay but maybe this is a speciality item.

I see that the sites mention 'carpet flange.' What is this and is it necessary? I assume that if I cut a square hole in the carpet that no support such as tack strips are needed around it. Or is this something worth doing to avoid wrinkles?

A carpet flange is a wider flange all around the box, or a wider separate flange you put under the box before setting it into the floor.
You DO need one!

Those prices, while high, are pretty near going rate. They are not cheap.

See the link. Do NOT mix brands. Get everything from the same Mfg.

Here is a link that might be useful: T&B carpet flange

Carlon makes a relatively inexpensive floor box that allows for a duplex outlet. These used to be available in the box stores, but I haven't seen them there in a couple of years.

Here is a link that might be useful: Carlon Residential Floor Box.

I see that both HD and Lowes carry a model. The Lamson circular 'box' is what I want but it comes with a hole saw, which I already have, which increases the price to $45 total. Definitely cheapest to use the big box stores for either a square or circular 'box'. I wouldn't mind buying a used version but don't see it on Ebay. Anyone know of other used versions?

I picked up a single outlet, rectangular box from HD. It is a Hubbell Floor Box Assembly. It comes with an old work metal box, outlet, gasket, and brass cover with single outlet screw in cap.

I originally tried the circular box / hole saw solution, but the plastic box mounts broke away from the brass cover after a few months (plugging in a device about once a week).

I can't find a link for it, but the UPC is 050169062364 .

Shadow, I was going to go for the Lamson circular box that comes with the hole saw. But I don't understand what you experienced. Are you saying the outlet itself came loose from the box from only the minor stress induced by plugging in and out a few dozen times? If so, this item sounds very shoddy. It might also explain the recall on it a couple of years ago when the plug was wired backwards. Anyway, I would appreciate understanding the problem you had with the circular box before I make a decision. Thanks!

The plastic cylinder where the wire connections are made is held to the brass floor plate with a few plastic tabs. After a short duration of light usage, the plastic tabs would no longer hold the plastic cylinder in place and it would flop around as you plugged/unplugged things.