Good Idea but never works.
We did that in Local Government. It ended up that if we wanted to buy something specialised we had to do the evaluation and negotiation ourselves, then the vendor would try to get on the supply list so we could actually procure it!
All well and good until other more generalised vendors got into price wars to get the preferred supplier option. This was obliviously at cost to something and if you weren't careful you had to buy the poor offering they were selling.
Meanwhile whilst with the Police, we noticed that every force had their own way of doing things complete with their own supplier and there was no way they were going to change.
Using the central supplier may help shortcut the procurement process, but what is really needed is a "Police team" consisting of the best workers, civilians, admin, audit, and even a few force managers, from multiple forces who have the capability to take the best available processes from each of the forces, tweaking them to suit every force and then amalgamate them into one system, easily paramatised to allow some local customisation, but in general just coming up with one system to put into each force. As it would be the same system, the data would be easily shared and manipulated with little effort as requirements grow.
The development team should be made up from proven in house programmers from the various forces and coopted specialists as required, making sure the expertise stays spread and in house.
It worked for us on a single force scale and we sold software on, but it would never happen now though! Crime Commissioners have to protect their empires.