Getting a letter of recommendation from university
(transforming the comments into an answer)

The point your question seems to be missing is that professors give recommendation letters, not institutions. LORs are "personal" commitments from people that know you, know how you work, behave, etc. People that do not know you will most likely decline or send worthless letters.

That counts old professors that remember you, regardless of where they are at the moment. Track them down. And there is not that much of turnover in universities, I graduated 10 years ago, most of my professor are still there... 

If no professor knows you, you won't get LORs. You don't need to "make friends" with the professor, good grades on his/her course should be enough. An undergrad research advisor would be better, but not necessary. It won't be a stellar letter, but for MS, it doesn't need to be. Just to reinforce: letters of recommendation are personal, not institutional.