How do we keep track of Eve on our Mobile Devices?

This is a bit of a tricky one, CCP closed Eve Gate at short notice sighting major concerns with the website.

After the discovery of a critical issue with EVE Gate last week, we’ve decided to bring forward the retirement of EVE Gate to close a vulnerability.
CCP Falcon: EVE Gate Shutdown – Wednesday September 27th

Falcon also stated that:

In addition to this, in the coming months we’ll be speaking with the EVE Community and the Council of Stellar Management to prioritize expansion of EVE Portal into a wider ranging companion app for EVE Online, so stay tuned for further information.

Getting to your question about how you get certain types of information out of game, the answer is the https://esi.tech.ccp.is/. ESI has an ever-growing list of read/write options for interacting with Tranquillity including a corporation history endpoint. https://esi.tech.ccp.is/latest/#!/Character/get_characters_character_id_corporationhistory

The big issue here is that you have to be able to either find someone who has developed a tool that you’re willing to use or make your own tool. I hope this helps.