If you are a person who avidly recycles, are you a recyclist?
It doesn't even show in the NGram Viewer but there's at least 77 books that mention "recyclist"
The OED (Oxford English Dictionary) classifies it as rare (and maybe it means something else too, but I'm not sure from the definition given), while the OALD and the NOAD don't even mention it.
The word you're looking for is recycler, which doesn't appear in the OALD, but on the NOAD it's given as a derivative word from Recycle and on the OED "One who or that which recycles (waste products, etc.)."
The third edition of the Oxford English Dictionary ("fully updated"online, March 2016) has the following definition for
An advocate of the recycling of waste products; a recycler.
and includes two usage examples not included in previous versions of the entry for this word, namely
1986 Peace News 23 May 17/1, I recommend all people who are tree lovers, recyclists and on low incomes to read friends' copies, buy copies collectively or go to the library.
1998 N.Y. Times 20 Sept. 26 Tap-water enthusiasts and recyclists worried that plastic containers will clog landfills.
The OED has also dropped the description rare for this word.
So, yes, it's a word, and it's a synonym for recycler. Not that we needed the OED to verify this.