Vikunja can be fully local. You just need a database (I think it only supports postgres). You can use the sqlite database for 1-2 users but I would use a full fledged database for an office setting.

I know they have a docker solution and if I remember correctly, a package solution as well. As others have said, you can get a RasPi or a cheap mini pc (better option in my opinion) and load Linux on it.

Once it is set up, you can use the machines ip to access it or put it in to a reverse proxy if you have one set up.

Not a direct answer to the last question, but you can run OnlyOffice in LAN environment. You can even use a domain.tld add ssl certificate to it and make it accessible only within your LAN. I did this about 6 months ago. Just saying.

My suggestion: go the cheap way to see if it works. Specially as you mentioned as it is for team work.

Buy a Raspberry Pi or a cheap Mini PC. Install Nextcloud on it. With Nextcloud you can install "Decks" what is their Kanban solution.

Try it out for some days. If it works, you can add further collab tools to the system.

I own a RaspberryPi 400 and it has been running 24/7 for over one year. I dont know how a SBC computer works when multiple ppl work with it, but as this is a super cheap solution, it does not hurt if it doesn't work.

I've run several Taiga instances on intranets for companies. Great kanban board.

Self hosted option is free

https://taiga.io/pricing-selfhosted

It looks like my favorite bug and task tool Jira can be selfhosted, if I'm reading it right: https://community.atlassian.com/t5/Jira-articles/How-to-run-Jira-in-a-docker-container/ba-p/752697