Is it OK to use abbreviations of the same university I am applying to?
The major issue with abbreviations is ambiguity. For example, OU could refer to Ohio University, University of Oklahoma, the Open University, and others. Other abbreviations like MIT are less ambiguous despite there being MITs in other countries. Similarly, longer abbreviations like UPenn are less ambiguous. Some universities, go by multiple names. For example University of Nottingham is often called Nottingham University, despite there being a Nottingham Trent University in the same city that goes by Trent.

When applying to a university, or for internal purposes, ambiguity is not an issue. If you are at, or applying to the Open University, in general everyone will assume OU means Open University and not Ohio University. In the rare cases where you are applying to the Open University from Ohio University, there could be ambiguity. Understanding the nuances of the name and abbreviation uses can be difficult. In general, you should probably steer away from abbreviations and just use the full name.