There are actually two aspects to take into consideration to answer your question :
- Find a DAW software that can be run on both the school's and your
- Access to the project file and audio files from both computer
The first point is the most tricky one:
Do the school computer and yours run the same OS. If not, is there a DAW software that can run on both ?
Do the school computer and yours have similar hardware performance (RAM, processor, etc.) ? Some DAW software requires powerful units to run flawlessly.
Does the DAW software license allow installation on several PCs ? Does it require a dongle that you can easily move from one PC to another ? Does it require third-party software to authorize ? Does it require network access to authorize ?
Does the DAW software requires administrative rights to be installed ? Does your school agree to users installing software on the school's PCs ?
Moving project from one PC to anther
Moving a DAW project implies moving the project file itself and the actual audio files.
Online storage: are the school's and your internet bandwidth fit to transfer large files ? What about the school's network policy ?
Movable storage: an external USB disk or USB key should be fine. Storage size to be determined by your project size (one hour of stereo audio files is around 600 MB).
With above points taken into account, potential candidates are (based on the idea that for this project, your budget is limited and you don't want to invest into full commercial licenses, and depending on your actual needs):
Pro Tools|First: WIN/MAC, free plus free online storage.
Audacity: WIN/MAC/LINUX, free and open-source.
Reaper: WIN/MAC/LINUX(experimental), full version free for 60 days evaluation.
Studio One Prime: WIN/MAC, free
Cubase Elements: WIN/MAC, 100 € approx. (as suggested by @rory-alsop).