I will take this occasion to point out that archery is often shot in the rain and supply some general suggestions as well as answer the question.
To answer the question directly, the answer is “no,” bows are mostly made out of anodized aluminum and plastic (the strings/cables are polyethylene) and are therefore waterproof. Wood bows are painted or varnished or waxed to repel water. Just be sure to take towels and whatever you need to dry off all of your equipment thoroughly, especially sights and release aids. If you just throw your gear into a bow case and zip it up, you are inviting rust on steel parts and even mold.
General Suggestions for Shooting in the Rain
I just got a tip on what to wear on your feet on rainy/wet grass days … waterproof golf shoes. These are made to comfortable, waterproof, and nonslip. Sounds like a good tip. Also, take extra dry socks. Swapping out wet socks for dry ones is refreshing and having dry shoes and socks to change into at the end of the day is very nice.
If you wear glasses, you’ll need a way to wipe them dry. If you use a scope on your sight, same thing. (Some use miniature cans of compressed air to blow them dry.) You’ll need a way to keep tabs and releases dry between shots (a belt pouch) and have a backup tab if yours gets drenched (as in the case in which you drop yours in a puddle). I use a lot of Baggies: one large one to tent over the arrows in my quiver to keep them dry and others to keep small parts dry. A quart size Baggie will hold all of the score sheets, keeping them dry between end scorings.
If you are going to wear a rain jacket, you want it to fit snuggly or carry a bunch of thick rubber bands so you can strap down the billowing fabric on your bow arm. Just putting your armguard on top of the jacket sleeve is often not good enough. You should test whether your jacket will be in the way before you need to use it in the rain. Some high-level archers use golf rainwear, because many of the rain jackets have removable sleeves.
The other thing about rain shooting, you probably will need to lower your scoring expectations, although when the FITA Round Compound world record score was shot, it was raining to beat Hell during the 90m end and later. Peter Elzinga still shot 1419 out of 1440. (He also set the WR for the 70m segment.) Most people, though, shoot poorer in the rain than they do in good weather. But everyone in the field is at the same disadvantage, so if you keep your attitude strong, you will have an advantage over those who get bummed out by the rain.
Good luck! (Luck, of course, is based upon hard work and preparation! “I’m a great believer in luck, and I find the harder I work the more I have of it.” Thomas Jefferson)