John Briggs listed a pub as his affiliation:
It is up to the journal, and most if not all journals will allow you to list "Astronomers for Planet Earth" as your affiliation.
People who are telling you outright not to do it, probably missed this part of your question:
"Most, but not all of us have institutional affiliations with various universities, but we would like to (also) use our grass-roots organisation as affiliation in our publications -- and we are wondering if we may do so."
For those without any other affiliation, the journal would probably ask more questions about their lack of affiliation than about the fact that the affiliation is "Astronomers for Planet Earth". Then if the people with no other affiliation are using "Astronomers for Planet Earth" as their affiliation and you are not, that would not seem right, so you should feel free to do it if you want to and the journal allows it (most likely, if you submit the paper with this affiliation, they will not even mention it in the slightest way, so you're probably worrying about nothing, but it's good that you're playing it safe and asking the community here).
Generally, institutions can get angry at you for putting their name on your paper if you weren't actually working there, but in the case where "Astronomers for Planet Earth" is someone's only affiliation and the organization wants you to list their name in the paper, your only obstacle would be whether or not the journal allows it and most likely they will. Some academics might not take you seriously if they haven't heard of your affiliation, and whether or not it's a good idea to do this will depend on many other factors, but whether or not you can do it is up to the journal and the answer is almost certainly a "yes".