I would definitely use "her family" since we want the reader to understand that it's her family that we are referring to. Using no article does not sound good and using "the family" is a bit impersonal as if the subject is not part of the family.

Family can be both countable and uncountable.

Olga_L, 'Family' is a collective noun. Collective nouns can take a singular verb or a plural verb. If you are thinking of 'family' as a unit then you can use a singular verb and treat it as singular noun. -My family is the best in the world. -Now that she works full time she doesn't have as much time to be with (her or the) family as she used to. If you use "the" it would be in reference to the particular family already UNDERSTOOD in context. Using 'her' would indicate her own family. Examples: 1) She comes from a very large family. Now that she works full time she doesn't have as much time to be with the family as she used to. 2) She worked part-time as a lawyer for the Jackson family. Now that she works full time she doesn't have as much time to be with the family as she used to 3) Now that she works full time she doesn't have as much time to be with her family as she used to. If you are thinking of family as a group of individuals you can use the plural verb and treat it as a plural noun. -My family are the best people in the world. -Now that she works full time she doesn't have as much time to be with family as she used to. This variation could only refer to her own family. There are no strict rules in the English language regarding the usage of collective nouns...and there are about 200 of them.

Formal = her or the Informal = no article

Could a native English speaker answer that? A Russian who was taught English would say her/the family, since we have been taught so, but I believe that "with family" is the best version at present.