The story mods are simply add-ons (extended endings) to the vanilla game, so they have no effect on gameplay until the vanilla part reaches its conclusion. The only difference is that at the end of each route, you'll be given the option to play the extended story. They'll just expand on the original tale and endings to include a few more happily ever afters, with the help of some Patreon exclusive images, sounds, etc. from Studio Kuma. You can install the mods from the start because there will be no confusion as to where the vanilla part ends and the mod begins.