Doesn't matter whether you're using Pro or basic LW

You should be able to add .zip files to your Public Shared list, so that your family can download them.

In LW go Tools > Options > My Files. Click on the Configure box, beside 'Configure unsafe sharing settings'. Now tick the box beside 'Allow me to search for & share programs with anyone' (if you click on the 'what are programs link', you'll see which file extensions you can share). Click ok. Click ok again on the main options page.