"My grandparents are farmers."
Translation:I miei nonni sono contadini.
The article must be omitted only for singular unmodified close family members: "nonni" is plural, so the article is needed. For the same reason it's "il mio nonnino" (diminutive, as affectionate form), and "il mio nonno inglese" (my English grandfather) as the noun is modified internally or externally.
There is no object here. "To be/essere" is a verb of state, not a verb of action. The predicate is describing the subject, it is not receiving the action of the verb.
And perhaps "don't need" was imprecise phrasing. The article is not used with professions when you use "essere", but it is used with professions when you use "fare".
I miei nonni sono contadini.
I miei nonni fanno i contadini.
Verbs of state, as in what is the status of the subject? They don't show any action. Nothing is happening. They simply describe the subject.
Some verbs can function as verbs of action or verbs of state. You need to pay attention to what comes after.
Alice is a teacher. (state--describes Alice)
Bob is tall. (state--describes Bob)
Carol looks tired. (state--describes Carol)
Carol looks at the wall. (action--something is being looked at)
Dave feels happy. (state--describes Dave)
Dave feels the desk. (action--something is being felt)
Eucalyptus smells like mint. (state--describes eucalyptus)
Edna smells the eucalyptus. (action--something is being smelled)
Frank seems calm. (state--describes Frank)
This is not a comprehensive list, but hopefully you get the idea.