I created an Elgg vagrant box with an early alpha of Elgg 2. The elgg version is a bit out of date, but it's easy to upgrade it from there since the config and setup is already taken care of.
[I don't think there are any linux images with recent versions of Elgg included available anymore. Some years ago there was one available somewhere but I don't remember where and I also wouldn't suggest using such an image. If you have set up a virtual machine with working LAMP stack it's really easy to install Elgg in it. And it's very likely you also want to test out your own selection of 3rd party plugins that are surely not included exactly as you require in any pre-built image.] --> Missed the posting of Matt
You wouldn't need an app to use Elgg. Elgg has an responsive theme and there are also 3rd party responsive themes available. So, you can access an Elgg site simply in a browser (maybe add a link to the start screen of your phone for convenience). I think there's one app available (useable with this plugin: https://elgg.org/plugins/1994332). But I have no idea what the app does and if it works without issues.
For an invitation only site you might want to take a look at this plugin: https://github.com/hypeJunction/Elgg-users_invite. It allows to invite users. I haven't used this plugin either and it seems you need some additional plugins to be able to use it (e.g. Friend request plugin available here on the site and some others maybe only on github). Elgg allows to set up your site as a private network (no one able to join). It also allows to invite people. But it does not allow to use these two features combined without some additional 3rd party plugins at the moment.
Sounds like SafeSocial:), with the exception of hosting arrangements. A note here, that if you are relying on a consumer internet access package, you will require truly unlimited super fast fibre broadband. You will also need to be aware that your own internet speeds will be greatly slowed, and of course, your server(s) will have to be constantly plugged into the mains electricity supply. And I hope you also have an expert available to manage and maintain your server(s). Good luck.
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