It's not technically incorrect, but this may depend on which side of the Atlantic you're on. See e.g.:
All grammar books that I have seen, from the 18th century to today, British and American, say much the same thing: that collective nouns can take a verb in the singular when seen as a unit, and a verb in the plural when seen as a group of individuals.
The difference between AmE and BrE is not so much a matter of technical correctness but of usage.
Americans have gone overwhelmingly for Formal Agreement (use of the singular), while we Brits have gone mainly for Notional Agreement (use of the plural).
But both are equally correct technically on both sides of the Atlantic, even if they may sound strange to one or other set of ears.
Which I (BrE) wrote (except with "don't) without even thinking. And it was accepted.
Could you say 'nie mają znajomych', if you meant the family members' individual friends? As opposed to meaning 'family friends'/friends of the family as a whole?
I think that it can be both: your family doesn't have common friends that come over a lot, and frankly, no one in your family has any friends at all.
What is the singular, nominative form of the word "znajomych"? I ask so that I know if it is feminine or neuter and know how the end of the word was needed.
"znajomy" (masculine acquaintance), "znajoma" (feminine acquaintance). It behaves just like an adjective. It can be an adjective as well.
In plural it's usually "znajomi" (masculine personal), because it seems quite probable that there are some men among one's acquaintances. But if a particular group of one's acquaintances consists of women-only, that's "znajome". As this sentence uses Genitive, you don't see the difference between those two.
I wrote "My family have no acquaintances", which means the same and is common (British) usage ,but it was rejected .