I can see where your IT team may not want you installing software on your PC without their approvals, but something like git (or SVN or CVS) should be deployed for all developers at every company (my opinion). Having no source control system at work while developing code is just asking for trouble. Especially when there are free, cloud based options (github) or inexpensive on-premise options. Probably even some free on-prem options too; I've never looked.
By not having any source control system at your workplace, you run the risk of changes breaking a critical tool and having no easy way to undo it. Or in the event of a disaster, how do you recover your code? Lets say your desktop hard disk (or SSD) bites the dust; how do you recover the source code changes?
If you are talking about developing code on your home PC and using GIT to pull it across the internet to your work PC, I think you can do this, but it is going to be complicated to set up as you would need to set up port forwarding on your router and have various ports open on your home network, plus you'd need to install tools, but there are a lot of tutorials on how to do that with a quick google. A MUCH easier option would be to take your source code from your home PC, dump it into a cloud git server (github, bitbucket, azure devops, gitlab, etc) and then pull it down from the cloud to your work PC. Just make sure to PUSH from both your home PC and work PC before you leave or you won't see your changes on the other computer.
I would work with your IT team to get git installed on your machine if you cannot install it yourself.
To solve the "single source of truth" problem that you have, git can definitely handle that, but you need a git server to push your git changes to and that server will hold the source of truth and all client machines will NEED to have git installed AND be able to access the server so they can pull the data down.