Short answer
Often yes, if your state law and local procurement rules allow it and the cooperative contract actually covers the products and services you need. A cooperative can save procurement time, but the agencies that win still run their own scripted demos and negotiate their own statement of work.
Cooperative contracts through groups like Sourcewell, OMNIA Partners or NASPO ValuePoint, along with state contracts, can let an agency skip a full RFP, and I understand the appeal when your procurement team is stretched. The first thing I would check is with your purchasing officer and attorney, to confirm that your state and your own ordinances allow piggybacking for this kind of purchase and that the underlying solicitation was competitive and covers both the software and the implementation services you plan to buy.
The bigger risk is that a cooperative contract can tempt an agency to skip the work that makes a selection successful, so even if you buy through one I would still define your future processes, compare at least two options against your own scenarios, and check references with agencies like yours. The cooperative sets a ceiling on price and some baseline terms, but it does not write your statement of work, and you can and should negotiate deliverables, acceptance criteria, key personnel and data terms on top of it.
Used that way, a cooperative is a sensible shortcut on paperwork rather than a shortcut on judgment. The Start at Zero Navigator covers the procurement options in more detail, and it is a free download.