Hi! In the library I work, we use Koha, which is probably the most well-known open-source library management system. This comes with the advantage of having a big community and having a lot of answers to questions you'll probably have, albeit the documentation is kind of all over the place. Just a heads-up, though: it only runs on Linux so, whoever is going to do the implementation must familiarize themselves with it if they haven't done so already. It's not a flawless system by any means but as far as open-source goes, it's the best and most mature.
There are a few demo servers you can try on their website: https://koha-community.org/demo/
The other open-source library management system I know of is FOLIO (their repo) but I haven't tried it or read much about it. I only know it's way younger than Koha (created 10 years ago, I think) and that EBSCO is one of its vendors. It may use newer technology but I honestly don't know. You can also try a demo server if you go to their wiki.
Hope it helped. If you have any questions, let me know :).