I have an opportunity to purchase, for cheap, a used kids bike. I don't know sizes, but it's small. Clown bike small. The tires are probably a foot in diameter. I would be buying it just for the tires. First of all, does that make sense @ $10? Or could I get new ones for a similar price? Also, are the tires too shallow to make a decent trailer that could carry groceries? I'm particularly interested in having something that can carry a 40lbs bag of dog food every month.

Sounds like 12 1/2" wheels. Should be fine. Price depends of supply and demand in your area, do not think you can find new ones same quality cheaper, but maybe free of charge in a dumpster or similar? Then you must maybe pay for fuel to go there.

Small wheels is not for offroad, but should be ok on flat areas. You could make one that would allow you to replace wheels later if you want or use different wheels for different jobs. Investegate for hub widths. My bamboo trailer has got this option. If I want to use the big box I use big 24" wheels. Low center of gravity is normally good.

I have an opportunity to purchase, for cheap, a used kids bike. I don't know sizes, but it's small. Clown bike small. The tires are probably a foot in diameter. I would be buying it just for the tires. First of all, does that make sense @ $10? Or could I get new ones for a similar price? Also, are the tires too shallow to make a decent trailer that could carry groceries? I'm particularly interested in having something that can carry a 40lbs bag of dog food every month.
I wouldn't. Many of these kinds of kids bikes are built very cheap, even down to the wheel bearings used. The spokes may be wire bent through the hub flange (two "spokes" are actually one long piece of wire) and welded to the rim.

I would advise to get the wheels someplace that sells bike and stroller wheels, so you can ask before buying that the bearings are serviceable--either normal bicycle ball bearings, or stroller cartridge bearings.
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20" steel wheels are not very expensive...been too long since I bought one to remember exactly how much.

With 20" you can buy a tube or a tire at any big-box store, and will have a wide selection of tires at any bike shop. Anything smaller gets hard to source spares.

Consider the value of your labor....does it really make sense to invest the work in building a trailer only to use junk parts?

Thanks for the advice. I don't have an LBS, so most wheel purchase options are going to be expensive. I am trying to do this on the cheap... but I may just have to wait until my next trip to civilization.

Thanks again!

Often the first one is not perfect so just start with what is awailable and make changes as you learn or find what you need. Many designs let you change the wheels if you decide to. I would newer buy heavy steel wheels, you can get second hand childrens bikes with alu rims for free or cheap. Many use them for folders, they are not all bad quality.

Maybe make a homebuilt "longtail" instead? Easy and really good solution.