Yes the Amazon Fire Stick does support Pure Flix.
I have an Amazon Fire Stick and have Pure Flix on it.
The Pure Flix channel requires a monthly or yearly subscription to use which is $10.99 per month or $99.99 per year.
I pay the yearly Pure Flix subscription because I plan to keep it for the long and longer but if you just want to check out Pure Flix on your Fire Stick you can pay monthly and save money.
You can also get a free Pure Flix 30 day trial to try it out to see if you like it before paying money.