This is one of the very few areas of basic grammar where US and GB English differ. In British English, collective nouns such as family ( or team, committee and so on) can be either grammatically singular or plural, depending on whether you're considering them as a single entity or a group of individuals. To my mind, only 'The Chung family will be travelling around Germany next week, won’t they?' sounds right. These are individuals, a mother, a father and children, perhaps, so it seems only logical to refer to these people as 'they'. Other members may disagree.

The first sentence is correct. We almost never refer to people using "it". The only exception is when we talk about the gender of babies. Example: She had a baby yesterday. It's a boy.

I'm submitting this answer to your follow up question as a separate answer, in order to keep the paragraph formatting: I would not use option number two. I imagine some US speakers would on rare occasion, but not often, since a family is not like a rock group, a company, or an army. A rock group functions as a single unit at specific times to produce music in front of the public, and a company or an army also works as a single unit to obtain their objectives. Examples: Pink Floyd is a great band, isn’t it? The army is really well organized, isn’t it? A family is usually not coordinated in the same way, and a vacation is not comparable to a military campaign or even a rock gig. Are we to believe that the entire Chung family is going to be traveling as a unit? It’s not even clear what one means by the “Chung” family — does that include aunts and uncles, in-laws, nieces and nephews, etc? That’s why I use singular in the main sentence (formal agreement) but switch to plural for the tag question. In real life, I would usually avoid the whole issue by re-phrasing it as: The Chungs are going to be traveling, aren’t they? I still think you’re smart, although I am starting to wonder how on earth you expect us to pronounce “bnpl” ;)

Hello Bnpl, that is a really interesting question, so here’s an answer from America :) (I’ll avoid quotation marks, since they don’t seem to be showing up correctly) As Su. Ki says, this is one of the few differences between US and UK usage. In the US, “family” is a singular noun (formal agreement), but even so, in your example, I would still use the plural pronoun and accompanying verb in the tag question and any other clauses that do not specifically use the word “family.” To make it clearer, I’ll use “to be going to” instead of “will.” The Chung family is going to be traveling around Germany next week, aren’t they? Note the singular “is” in the first clause, but the plural “they” in the second. The same applies to rock groups, for example. “U2 is going to be here next week, aren’t they?” I would also use the plural possessive: The Chung family is washing their hands in the kitchen. This usage may be considered similar to “Someone is washing their hands in the kitchen.” On the other hand, if we are truly thinking of the family or band as a single entity, the tag question would indeed be neuter singular: “U2 is a great group, isn’t it?” (Their greatness is as a single group — I’m not commenting on any of the members as individuals.)

I think the first variant is correct.