There is a (weaker) version of weak compactness which does not imply strong inacessibility: just ask that the infinitary language allowing conjunctions and disjunctions of $< \kappa$ sentences is $( \kappa, \kappa)$-compact (this means that every set of $\kappa$-many sentences is satisfiable iff every subset of $< \kappa$ sentences is satisfiable).
The continuum cannot be weakly compact even in this weaker sense, by a characterizability argument: consider a model with a definable bijection which assigns to each element of $\kappa$ a function from $\omega$ to $\omega$. If this is a bijection, the $L _{\omega, {\kappa, \kappa}$-theory omega}$-theory of this model implies that you cannot add new elements to$\omega$, hence no new element to$\kappa$(this would give a new function from$\omega$to$\omega$, which is impossible). This contradicts$( \kappa, \kappa)$-compactness. However, what is interesting is that you can have$2^\omega > \kappa$in the above situation, that is, you can have "weak compactness (in the weaker sense) without strong inaccessibility". This is the main result of an old paper by W. Boos: Boos, William, Infinitary compactness without strong inaccessibility. J. Symbolic Logic 41 (1976), no. 1, 33-38. Even if the continuum cannot be weakly compact, there are interesting connected results. 1 There is a (weaker) version of weak compactness which does not imply strong inacessibility: just ask that the infinitary language allowing conjunctions and disjunctions of$< \kappa$sentences is$( \kappa, \kappa)$-compact (this means that every set of$\kappa$-many sentences is satisfiable iff every subset of$< \kappa$sentences is satisfiable). The continuum cannot be weakly compact even in this weaker sense, by a characterizability argument: consider a model with a definable bijection which assigns to each element of$\kappa$a function from$\omega$to$\omega$. If this is a bijection, the$L _{\omega, \kappa}$-theory of this model implies that you cannot add new elements to$\omega$, hence no new element to$\kappa$(this would give a new function from$\omega$to$\omega$, which is impossible). This contradicts$( \kappa, \kappa)$-compactness. However, what is interesting is that you can have$2^\omega > \kappa\$ in the above situation, that is, you can have "weak compactness (in the weaker sense) without strong inaccessibility". This is the main result of an old paper by W. Boos: Boos, William, Infinitary compactness without strong inaccessibility. J. Symbolic Logic 41 (1976), no. 1, 33-38.