Creating such institute might hit some problems about local laws that do not accept/foresaw those things or maybe create some obstacles for possible members. Many countries feature laws that would prevent that such institute could ever be created because in order to allowing operation and/or recognizion as an educational and/or research institute, they demand things that only makes sense to physically existing ones. You can't circumvent that, because trying to do so would imply that your institute would be operating illegally, which would be a very bad idea.
In face of that problem, you might need to research what would be the best country in which the institute would be based to ensure the minimal possibility of such problems.
Further, many courses such as chemistry, medicine or geology needs quite specilized labs or practice lessons that will demand that at least some of the institution courses will need to have physical instalations.
But, afterall, I can assure you that virtual institutes do exist, and there is a lot of them. In Brazil, there is a lot of universities out there that provides many virtual graduation courses and many strictu sensu specialist virtual courses. Some institutes features only virtual courses at all (so you might call'em "virtual institutes"). However, there is no virtual master or PhD courses here yet, and I actually searched for that (dunno the reason though).
In fact, anedoctally, I was thinking about joining a strictu sensu virtual specialist course, but in the end I opted for a traditional one. My girlfriend, however, just started her virtual strictu sensu specialist course. In such courses, lessons are video streamed, homeworks are delivered via e-mail or via upload somewhere, students and teachers talk through course-specific forums, e-mails or some form of chat. With that, tools like Moodle and Skype are omnipresent. Everything is over the internet.
Those online courses features recognition from the government as educational institutes and the diplomas that they would eventually emit for their students is recognized by the government and by other universities as valid. In fact such online graduation and specialist courses are popping-up at such fast pace, that in a few years, it is likely that they will dominate over traditional ones and be the default style. For those courses that can't have 100% virtual lessons, they would become partly/mostly virtual, with only the parts that can't be virtual being not virtual.
Virtual courses have many advantages, such as the students being able to watch the lessons in their preferred times freeing them up to be able to work and earn his/her money through the day. This also free up the teachers to be able to make their best schedules. Also, since lessons are normally video streamed over the internet and NOT live, students may pause or rewatch them just as they would do in a YouTube video, which makes they more confortable to watch (you could take lessons in your bed watching them on your smartphone). This also eliminates those problems that may make some students lose lessons by not being able to go to the class in a particular day due to some particular problem or by losing part of the lesson due to arriving late because there was a traffic jam, or the problem of losing or not understanding an important part of the professsor's talk (in presential lessons this might be a killer, but in video lessons this is no problem since the student can just rewatch that part of the video). Further, since this is not a live video, if the teacher makes some mistake or get confused somewhere, he/she will just rerecord that part instead of confusing his/her students. Another advantage is that people living in areas where there is no traditional course in their area of interest (as is the case of many smaller towns) do not need to move to somewhere else in order to eventually take his/her lessons. Also, by being online, institutions can cut a lot of costs, which also lowers the cost of education for the students. This also allows for larger classes, lowering the needed rate of teachers over students in order to operate, and allowing more students to take courses. There are some disadvantages though, as student-teacher and student-student relations are severely degraded and some students aren't productive taking lessons that way, which makes them go back to traditional presential courses. Also, many types of reasearch becames harder or slower to do. And of course, the student needs the access of a reliable internet connection at least during part of the day.
Anyway, even institutions that features fully-online courses or mostly-online surely exist, they still must feature a physical location (even if that happens to be just an office somewhere) as a requisite to be recognized by the government (at least in Brazil, maybe some other countries do not even need that). Further, in order to be allowed by the government to operate and be recognized as courses, they must feature presential exams, even if everything else is online. Many courses also features semestral or annual student meetings.
So, virtual institutes surely exist, and they are very active doing real research and providing real lessons. All over the internet.
What about scammy institutes? I can assure that they also surely exists, sadly. Many of those virtual institutes are honest, but some aren't (to be fair, some traditional ones aren't too), and many of those institutes are just façades for making easy money by selling out undeserved diplomas. They features very low quality lessons and very stupid exams, if those aren't simply faked altogether. The government are always watching for those and frequently closes those scammy/dubious courses/institutes, but many of them are able to live very long without being caught by government course quality measurements (it is not very hard to trick out the government). With that, many online courses are still seen negatively by many people, to the point that some people simply refuse to hire people who took online courses, which in turn also makes the life of the honest online institutes and students harder.
Further, there is several grades of "scammyness". There are institutes that do dubious/scammy practices only in some courses or some disciplines or just with some students. For example, they might feature honest courses, but might also secretly/illegally sell diplomas and fake scholar records for students that pay enough cash for that without ever watching a single lesson (of course, both parts will never admit that in public, so you will never see that service being advertised). Also, some courses might selectively be too easy or too hard depending on the student (for example, those who pays regularly get easy exams, those who delay or default payments get hard exams). Some may secretly pass a failed exam and fake the grade if the student bribe them for that. There are many other ways where institutes might be in the category "not fully honest but also not fully scammy either". However, those are old practices from already existing unethical traditional institutes that just showed up their already seen face and vigour when they went to the virtual space.