You don’t need a licence, it’s all free and open source software. For hosting it on your own web server (like a blog / website, but with a bigger server) it would cost 30-50 EUR/month (rough guess). However for the first time when the project does not have much budget or tech people, there is a much simpler hosted solution (= “on other people’s server”), also based on OpenStreetMap (and it can be changed to the self-hosted map software later without losing data).
For that, I propose you use OSM uMap, in a way similar to my instructions for disaster response mapping. Just that you don’t map a disaster, but businesses etc. in your city. You can then embed or link this map on your own OurGhema website and other websites.