Is it grammatically correct - "Her and her looney boyfriend nicked my truck of cigarettes and drove it in the river."

Her and her boyfriend used as the compound subject of a verb is "substandard" though not uncommon.

The grammar rule is that subjects are normally in the nominative case (she) not the objective case (her).

We don't say "Him likes bacon" or "Her likes bacon" but "He likes bacon" and "She likes bacon".

The fact that the subject is compound should not change the case of the third person pronoun from she to her.

Speakers also make the other mistake, using the nominative when they should use the objective:

*Bill came along with her and I to the movies.

*Finding an apartment in our price range is difficult for my girlfriend and I.

In his well known 1964 article Negation in English, Edward Klima characterized the use of the subject forms in contemporary English this way: the subject form is used for the unconjoined subject of an explicit finite verb, but otherwise the object forms are used. And I think that's right. Of course, there is an old-fashioned archaic style in which you'd use "she" instead of "her". Your example sentence sounds fine to me, but if you wanted to sound stuffy, you'd use "she".

It occurs frequently in vernacular English, it is by many authorities considered to be inappropriate for formal English.

Even though it does occur in vernacular English, I'd say that there is hardly ever a good reason to not use “she”, as the subject pronoun will be accepted by all speakers, but some speakers reject the object pronoun, and almost no speaker would consider it pompous unlike with, say, “It was she.” opposed to “It was her.” which would be considered pompous by a certain number of speakers, though some speakers would also consider “It was her.” to be incorrect, however common.

A good reason to use “her” would be in fiction, to accurately portray vernacular language.