http://www.refugees-welcome.org.uk/refugeeswelcome-groups-training-action/ has a list of local groups with email contacts, so you can try the one nearest you.

https://do-it.org/ is a great idea in theory. If you click more options, there's a "do it from home" filter.

Volunteering to be a trustee can sometimes be a good option, if the charity is happy to have you dial in to a conference call for meetings, or if you can get there in person occasionally (usually once every three months for about 2 hours). Then you answer emails in between meetings.