Chewie,
"Cheap" and "Quality" rarely go together in the world of recording, assuming you want a decent outcome. You need to keep that in mind throughout your design and assembly process.
4'x4' is not a large booth, but it's doable assuming your singers aren't pasta lovers
I'd like to submit an alternative to building a miniscule vocal booth. Think about this for a while...
You have a small studio as it is, cutting it down even smaller to make a vocal booth is only going to cut into your existing space, and possibly significantly alter the sound within the studio as compared to how it is now.
What you can do, instead of building a booth, is build, or purchase, isolation panels that are free standing. You can move them around to different configurations, i.e. three panels in a "U" shape for a vocalist, or more of an open "C" shape for something larger, like a drum kit. Much more versitle than a constructed booth with the limitations I percieve you to have. The other advantage of being free standing, should you ever move, or take over another space within this house, you can simply carry them with you to your new space!
Two layers of 3/4" MDF glued and screwed together, with a thick layer of rigid fiberglass installed to one side, with fabric stretched over the fiberglass and stapled to the back. Not very expensive to make, and useful for more than one purpose.
Now, that you have isolation panels in your studio, do know that they aren't going to eliminate all sound and reflections around the room. Just enough that your microphones shouldn't pick up anything but the performing artist. As a console operator, you'll need to recording your artist using headphones, so your monitors don't bleed back to the microphones. Also, record the artist dry, so you get the nicest possible recording you can, and add effects and EQ during the mix, as during mixing you can use your monitors since no one is singing at that point.
Over the years I did many voice-over recordings this way, and the results were "above average" to "excellent", and didn't take up a lot of space. I leaned them all against one wall that wasn't covered in gear, when I lived in my old, much smaller, place.