Do you mean being paid a salary to be an (independent) astronomer, or having telescope time, travel, etc. funded so you can do research?
I completed a PhD in astronomy but left when accepting a postdoc position would have meant moving my family across the country and the 'day job' is now in IT. However, I'm still an active research astronomer - this year I had two VLT proposals accepted as a PI (roughly 120k Euros in telescope time), and was co-investigator on four more accepted ESO proposals. I'll publish roughly a dozen papers (including four as first author, three as second), will speak at conferences and last year had an ESO press release on my work which was picked up worldwide. So it's perfectly possible to continue in top-level research without a faculty position - plus I have the advantage of not teaching :)
I apply for telescope time through the usual routes, e.g.
and many sources exist for conference fees, travel, expenses, etc., e.g. amongst others I use
(I'm UK-based, which explains those choices.) But if you want something to pay a salary as well then I think options are much more limited. There are a few specialist postdoc support programmes, e.g. for returning to the subject following a career break, but options for ongoing funding are very limited - and would be very competitive. I know a few people who make a living writing for astronomy magazines, etc. alongside their research, but the majority of 'non-faculty' researchers like me that I know of have a traditional 'day job' that pays the bills and is flexible enough to allow high-level research as well.
EDIT: One other avenue of investigation are fellowships from the likes of the Leverhume trust (http://www.leverhulme.ac.uk/funding/ECF/ECF.cfm or http://www.leverhulme.ac.uk/funding/RF/RF.cfm). These are linked to the traditional PhD -> postdoc -> Faculty route, but are somewhat different to the traditional postdoc position. Royal Society fellowships also provide five- or ten-year salary funding (http://royalsociety.org/grants/schemes/university-research/). Again, these are UK options and I'm unsure where you are located, but these would be options I'd very seriously consider if I was ever going to try to return full-time to academic life.