My institution has access to a very limited selection of ebooks published by Springer-Verlag. Is it possible to buy individual access to their whole collection myself? I'm a PhD student in case that changes anything.
You must ask Springer; they are the only ones who will know. There's a good chance they will be okay with you purchasing such a subscription since after all it doesn't cost them much to sell their collection to one more viewer.
However: although individual subscriptions tend to be way cheaper than institutional subscriptions, you are talking about subscribing to the entire Springer collection. This is a veritable mountain of information! Checking the Springer website right now, they have a collection of 301,765 books, and a publisher of their scale is going to be publishing several new books every day. Small wonder it is going to break your wallet to subscribe! Further, most of them won't be at all relevant to you and you won't have time to read them anyway.
If you can't get institutional access to this collection, you are probably better off buying each item you need one by one. Even then, be sure to ask your institution's library - they might be able to acquire/rent copies of what you need for cheap or even for free.
Even if it was technically possible, it's likely to be ridiculously expensive. Big consortiums of libraries have been struggling with Springer for years about the price of their subscriptions:
If this kind of academic groups which have very signifcant budgets and teams of lawyers to defend their interests have difficulties paying the fees, it's not hard to understand why there is no option for an individual to subscribe to the full service. And this is certainly the reason why your institution has only access to a limited selection of publications.
So as far as I know it's not possible for an individual to subscribe (unless they are a billionaire).