While I don't know much about beet harvests, there are fruit and vegetable harvests all over the country.
Strawberries, Cherries, Apples, Pears, Peaches, and many many more. Many vandwellers follow the harvests and make more than enough money to see themselves through the winter.
I know vandwellers who do yard care during the summer then head south for the winter. Some even own their own lawn care businesses and return every spring to the same town to run them.
Temp agencies and day worker or day labor centers can also be great resources for people living on the road.
If you know how to make money at them, there's flea markets and swap meets all over the place.
One popular option for full timers is to make money online, lots of opportunities there.
Then there's businesses you can own and operate from almost anywhere, like painting house numbers on curbs, or any one of dozens and dozens of others.
Busking (Street corner entertaining) for donations. Some people report making upwards of $100/hr doing this.
There's an almost unlimited number of businesses that can be started for free or on a shoestring budget, that can be run from almost anywhere. You are only limited by your own imagination and creativity.
Opportunity is everywhere once you learn how to recognize it, and take advantage of it. Hard labor, easy labor, mental labor, fun labor, it's all out there just waiting for you. Find something you love, and you'll never have to work a day in your life.