We derive new Hanson-Wright-type inequalities tailored to quadratic forms of random vectors with sparse independent components. Specifically, we consider cases where the components of the random vector are sparse α-subexponential random variables with $\alpha>0$. When $\alpha=\infty$, these inequalities can be seen as quadratic generalizations of the classical Bernstein and Bennett inequalities for sparse bounded random vectors. We further apply the inequality for a local law and complete eigenvector delocalization for sparse α-subexponential Hermitian random matrices.