Which one is correct and why?

Solution 1:

A. Across the Chesapeake Bay from the rest of the state, there lies Maryland's Eastern Shore, whose farms produce beans, tomatoes, and other garden vegetables.

This uses 'existential there' whose use is not ungrammatical here, but which does give a rather archaic flavour in this instance.

B is correct (if one accepts the capitalisation), and the inversion after the adverbial is usually preferred.

C would need a fair overhaul:

Across the Chesapeake Bay from the rest of the state – Maryland's Eastern Shore lies there. The farms of that area produce beans, tomatoes, and other garden vegetables.

Notice that this 'there' is the locative one. This version sounds rather poetical.

D probably requires a slight re-ordering or (better) re-structuring:

Across the Chesapeake Bay from the rest of the state, Maryland's Eastern Shore, whose farms produce beans, tomatoes, and other garden vegetables, lies.

Across the Chesapeake Bay from the rest of the state, Maryland's Eastern Shore lies. Its farms produce beans, tomatoes, and other garden vegetables.

Solution 2:

A is grammatical. There, in the OED’s words, is ‘used unemphatically to introduce a sentence or clause in which, for the sake of emphasis or preparing the hearer, the verb comes before its subject’.

B dispenses with there, but it is equally grammatical. Maryland's Eastern Shore is the subject and lies is the verb.

C is ungrammatical. That is because there, when it is used as in A, must normally precede the verb.

D is just about grammatical, but is unsatisfactory, because it separates the modifying relative clause (whose farms . . .) from its antecedent.