Short answer
Write scripted demos built on your own scenarios and sample data, send them to vendors ahead of time, and have the people who do the work score each scenario as it happens. The vendors that win your demo should be the ones that can actually perform your work, not the ones with the most polished presentation.
Every vendor has a demo that makes their product look wonderful, and if you let them run it you will spend two days watching the parts of the system that work best and none of the parts that matter most to you. The fix is a script, which in my experience should be built from the ten or fifteen scenarios that define your agency, things like processing a grant reimbursement, handling a retroactive pay change for a union group, or amending the budget mid-year, with enough detail and sample data that the vendor has to show the real steps rather than a slide.
I like to send the scripts to every vendor a few weeks ahead so nobody can claim they were surprised, and then I put the evaluators who actually do the work in the room with a simple scoring sheet for each scenario, because your payroll lead will spot a clumsy workflow that an executive will never notice. It also helps to reserve time for an unscripted round where your team asks the vendor to do something on the spot, since that is where you find out whether the presenter knows the product or just the script.
After the demos, compare scores while memories are fresh and write down why each score was given, which gives you a defensible record if a protest ever comes up. The Start at Zero Navigator includes guidance on building demo scripts, and it is a free download.