Using Google Chrome on macOS to run Windows executable (.EXE files)
No, you can't run exe's via Google Chrome. You may be able to use some sort of remote desktop solution such as Chrome Remote Desktop, whereby you install the exe on a Windows computer which you can see and control from your Mac—but that's not quite the same thing.

There are a number of solutions for running Windows executables on a Mac, each with their own set of compromises:


Wineskin / Wine. Works perfectly with a minority of Windows programs, but doesn't work at all with others. When it does work, this is the best solution by far.
Bootcamp Assistant, which is preinstalled on every Intel Mac in the Applications → Utilities folder, will guide you through the process of installing Windows on a separate partition. Once done, you will be able to reboot your computer into Windows—effectively turning it into a PC, temporarily—and reboot again to return to macOS. Note that this will require a purchased copy of Windows. While rebooting is a pain, this is the most "no compromise" solution, in that every Windows app will work and you won't loose any performance.
Virtualization software such as Virtualbox (free) or Parallels (paid, and expensive). Like Bootcamp, but you'll be able to switch to Windows without rebooting your computer. The catch is that you lose performance—applications will run considerably more slowly than normal. Note that the performance loss is much worse in Virtualbox than in Parallels—you get what you pay for, in this respect.


The problem is that all of these solutions requires either additional third party software (1, 3) or administrator privileges (2). I don't know your situation, but it seems to me that you need to have a discussion with a supervisor or similar about obtaining the resources and privileges necessary to do your job.