You can plant in any container which is deeper than about 8-10 inches and has drainage. Even 6 inches is OK for shallow rooted plants like lettuce or radishes and onions.
Zuchinni get a good root system happening, so a deep tub for example, may give better results.
Tomatoes are perfect for a bucket and for the area they take up they score pretty high in the returns too. You could even have some cherry tomatoes in a bucket hanging from a hook to use your vertical space. A trellis for some cucumbers will do a similar job.
Onions would be OK in a polystyrene tray type box like they have at fruit shops as would radishes, garlic and lettuce. Using "pick and come again" lettuces like cos lettuce will give good continuity of supply. Some silverbeet gives good return for a small space too.
Carrots grow well in buckets.
You need a minimum of 6 hrs sunlight to get these vegies thriving.
So in answer to your query, yes, you can grow this stuff in buckets and it's not only the price but the taste which is superior as well as the satisfaction that you grew your own.
Some devoted container vegie growers on here. Stay tuned and welcome to the forum.

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The secret is the soil.

Crystal, you're gonna have fun with your project !
I'm a container gardener too, and it gives me alot of joy.
Be sure to invest in some good potting soil. I use both E.B. Stone and Miracle Gro soils, whatever's on sale.
Good luck and have fun !

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