Can I show my Instagram posts on my store home page?

Yes, with the Instagram Gallery section (in the social category under + Add). It isn't a live feed, so you choose the photos yourself:

1. Fill in Heading and Instagram Handle. The handle shows as a link to your profile.
2. Optionally add a Profile Link (optional).
3. Under Tiles, click + Add tile for each photo and set Image, Post Link (optional) and Caption (for screen readers).
4. Set Items per View and, on the Style tab, Tile Gap.
5. Click Save.

New Instagram posts don't appear by themselves, so swap the tiles when you want fresh photos. Lookbook Instagram Grid is another layout for the same idea.