Do I need to install the app for every theme on my store?

No, the app is installed for all themes.

It is possible (but not probable) that the app does not work on an unpublished theme if you're using the Shopify theme preview mode, but it should work once you publish the theme.

In case any issues are faced after changing the theme you just need to contact us and we will fix any integration issue for you as soon as possible.