Possesive 's

Yes, it is correct. Alternatively, you could use "The Berg Family Kitchen."

Thanks!

Cheers

In the interest of full disclosure, i should point out that some people would refer to the Berg Family as the Bergs, in which case, one would write Bergs' Family Kitchen (note the placement of the apostrophe in that case). One could also write Berg Family's Kitchen, or The Berg Family's Kitchen. Isn't the English language wonderfully silly?

They themselves are, of course, the Bergs' Kitchen's Family.

I heard they are moving to Kitchener, Ontario.

Cheer's.