Yes, this can, for example, be checked using fppf-descent.

Pick a presentation $p:U\to X$ and pull-back everything along $p$. Since blowing-up commutes with flat base change, you may then use the universal property for $Z\times_X U\to U$ and $\tilde{X}\times_X U\to U$ to deduce that if the inverse image of the ideal sheaf to $Z$ is invertible, then there is a unique lifting $Z\times_X U\to \tilde{X}\times_X U$. Similarly, there is a unique lifting $Z\times_X R\to \tilde{X}\times_X R$ where $R=U\times_X U$. Then you conclude that there is a unique lift $Z\to \tilde{X}$ by fppf descent.

[Edit: I misread the question as asking for $X$ to be an algebraic stack. If $Z$ is an algebraic stack (and $X$ is a scheme or an algebraic stack), then you can similarly pick a presentation $q:V\to Z$ and deduce that there are unique morphisms $V\to \tilde{X}$ and $V\times_X V\to \tilde{X}$ and hence, by fppf-descent, a unique morphism $Z\to \tilde{X}$.]