Disc Golf can be played anytime. Most people are fair weather players but there are many people who play after the sun has set and in rain and snow.
Most of the major disc manufacturers also make their most popular discs in glow plastic. These discs are made specifically to be played in the dark. They glow in the dark after being exposed to enough light. Instead of buying a second set of discs, you can also sticker glow tape onto a disc or LED lights onto a disc made of translucent plastic.
Discs made out of softer plastics are grippier in rain and snow. For rainy weather play, a rainfly for your bag, microfiber towels, raingear, waterproof footwear, gaiters, and a golf umbrella will definitely help. For snowy conditions, tape long streamers (like ribbon for wrapping presents) onto your disc so that you can look for the streamer if your disc slides under the snow. Just remind yourself that less than ideal weather probably means you won't play your best round so you should be playing just to enjoy the game.
Tip: If you like to play early in the woods, try to walk behind another player if you don't want to be the one to break the cobwebs.