The question of Moodle being available on mobile devices is not so much a technical one, as what the user's expectations are. An institution can switch on access via the Moodle App, provided they have a later version of Moodle. Also they can provide a responsive Moodle theme. But institutions have invested years of work in the classic Moodle look and are understandably cautious in making a change
Users will have expectations with an App or a Mobile interface that everything will be different, but underneath it is the same Moodle and same course content. The result can be frustration and so I understand why institutions are moving slowly.
As a result of being a student in a course on mobile educational design I have been confident enough to volunteer to be a pilot user for a responsive Moodle theme for my course ICT Sustainability, starting February 2016 at ANU.
In a way Moodle's success is a problem for transition to mobile devices. Hu, Lei, Li, Iseli-Chan, Siu and Chu (p. 5, 2015) report that students with two or more years experience with Moodle were less likely to try the mobile version, than those with less experience. Also students who considered themselves as having limited IT competency used the mobile version more.
Hu, Lei, Li, Iseli-Chan, Siu and Chu (p. 9, 2015) conclude that students did not prefer Moodle on a mobile phone, but would use it when necessary. I suspect the same would apply to the Moodle App (although that was not tested). Institutions are therefore right to be cautious about introducing the mobile interface, as it is likely to create expectations it can't satisfy. Just adding a mobile interface doesn't make for mobile learning.