What vegetables grow best in raised container gardens in back yard?

What is your growing zone?

There is so much more than what grows in a raised bed. Soil, sun, nutrients, water...vegetables need minimum of 6 hours of sun a day.

Contact your County Ag. extension Office. Since they are local, they can guide you best. If not, go to a plant nursery - not one at a big big box store, they can assist you too.

tomatoes, cucumbers, herbs, carrots, eggplant

You can grow most any vegetable well in a raised bed. Depending on how deep it is and where it is located has as much if not more to do with which vegetables you should plant and how well they will do in the bed. I grow all my garden plants in raised bed and they do well.

Do note that a raised bed requires you keep better track of watering as the water drains to the bottom of the bed. Good to pull the roots down further into the soil and support your plant but bad if you don't watch and make sure it doesn't dry out to much.

Here's some ideas

https://www.hometalk.com/diy/grow/plants/5-best-container-vegetables-for-beginning-gardeners-3400506

These tips might help: https://www.google.com/amp/s/www.almanac.com/content/raised-bed-gardens-and-small-plots%3famp