I think there is a point not enough clear in your question...
The most familiar logics in the modal family are constructed from a weak logic called K (after Saul Kripke). [...] A variety of different systems may be developed for such logics using K as a foundation. [...]
K results from adding the following to the principles of propositional logic [...] [emphasis added].
The last statement means that the axioms and rules (and meta-logical concepts) of propositional logic still apply, like e.g. modus ponens.
In particular, this means that in system K we can use the concept of propositional tautology as well as the "standard" completeness of the propositional part with regard to the truth functional semantics.
All this long premise to say that a formula ¬α ∨ α of K is still a tautology, and thus provable, for α whatever.
Thus, substituting with Lp for α we get ¬Lp ∨ Lp.
The reason is simple: if Lp is true, then clearly ¬Lp is false, and thus, by truth table for ∨, ¬Lp ∨ Lp is true, and the same with Lp false.
For a detailed proof, see e.g. Edward Zalta, Basic Concepts in Modal Logic (1995): Ch.3.2 Tautologies are Valid.