Currently my team uses Team Foundation Version Control (TFVC), with our backups stored via a Team Foundation Server SQL Express database. We're starting to run out of space (since Express is limited) and I wonder if we used Git for source control if we would use less space?
Impossible to say. Git and TFVC are fundamentally different version control systems with different considerations around what should and should not be stored in the repository. As an example, Git is very efficient when dealing with text (such as source code files) and very inefficient when dealing with binaries.
The solution here is not to switch source control systems, but rather to upgrade to a non-Express version of TFS or to migrate to VSTS, either of which would solve the actual problem, which is that you're bumping up against Express limitations.