Thank you for your post. This is such a lovely idea and so generous of you.
I've had a search too and haven't found anything specific, but I think part of this is because there's not enough people like yourself who want to commit the time and energy for this.
I wonder if it might be worth finding an organisation that resonates with you and giving them a call or sending an email to see if they would like some Christmas cards. Nursing homes can absolutely be an option, but you can also try places like hospitals or respite (asylum seekers/homeless shelters).
Whatever you decide to do, I'm sure they will be extremely grateful. Feel free to let us know how you get on.