One wallpaper to rull them all...
1) Check visually if everyone is using a Widescreen monitor. If a lot of users are still using some oldie models this will be a 4:3 proportion. So you need at least 2 versions.
2) Make a small census of the specifications people are using right now. Right click on your background and press Screen resolution. Probably not everyone are using Full HD, some laptops use the same proportion but not at that high reslution.
I would start a wallpaper Full HD, 1920x1080 px, and crop it and resample it if I need a 4:3 image, lets say 1280x960 px.
3) Edited. There is a new player in town. 21:9 proportion called WQHD, This is an Ultra wide screen. Some samsing models are 3440x1440px.
4) When configuring it use the Adjust option so the wallpaper resamples to fit the resolution. If you think some computers deform it or make it a little blurry, do a version of that native resolution. But I think with the 2 options I mentioned you are ok.
Although there are a lot of phisical proportions, the final decision if you need to make all of them is if you have very noticable features. A human face, a perfect circle as the main design.
Here is a good reference of the screen resolutions and proportions. https://commons.wikimedia.org/wiki/File:Vector_Video_Standards2.svg
(Note where the 16:9 and 4:3 proportions are positioned in relationship of the others, that is why I would use thoose two as a base)
Far more important than being picky on the screen resolution is that you have a great and functional background design. I can have a perfect fit background on my computer, but if it is ugly, stressfull, produce flickering, make my eyes stress, etc. It won't work.
If the design is great, and comfortable, well, I can live with a small strecching on it.
Our graphics design guy left
I would worry a bit on that. Hire a good designer to have a great image. Not everyone will love it but not everyone would hate it.