To a positive-definite even lattice $Q$, one can associate the lattice vertex
algebra $V_Q$, and any automorphism $\sigma$ of $Q$ lifts to an automorphism of
$V_Q$. In this paper, we investigate the orbifold vertex algebra $V_Q^\sigma$,
which consists of the elements of $V_Q$ fixed under $\sigma$, in the case when
$\sigma$ has prime order. We describe explicitly the irreducible
$V_Q^\sigma$-modules, compute their characters, and determine the modular
transformations of characters. As an application, we find the asymptotic and
quantum dimensions of all irreducible $V_Q^\sigma$-modules. We consider in
detail the cases when the order of $\sigma$ is $2$ or $3$, as well as the case
of permutation orbifolds.

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