A discussion here, in which some Instagram alternatives (from the perspective of service) were discussed:
Building your own Instagram in RapidWeaver is likely to be more challenging and you would have to make some compromises as to how it functioned. I would personally consider Droplet + ProGallery in combination or use the WebYep Gallery to upload images directly into a lightbox gallery with captions.
Both solutions would require some image editing on the part of the user - to reduce image sizes and apply the Instagram filters they want. But it terms of uploading, it would work anywhere, without dependencies on needing apps.