There is a perception that friction fire prefers dry conditions. Admittedly it is far easier in dry conditions and not all woods like dampness and I can't use the hand drill with damp wood and I'd prefer the dry any day! However, It can be possible with some wood species which are damp using the bow drill. I'm not saying you can do it with soaking wet, soggy all the way through wood but it can sometimes be possible with damp wood as I've tested recently. Of course, if it's chucking it down with rain you need to get into a sheltered spot and of course you need dry tinder and wood. Personally I prefer to carry everything I need with me but if you have the knowledge, it can be possible but a bit more prep may be needed (e.g. making feather sticks etc, using tinder fungus.) It's best to try and find dead standing wood with bark on as rain runs off it and it doesn't get as wet as wood on the ground or horizontal wood.
As an experiment I soaked a freshly made hazel drill (without bark) and an Ash hearthboard in water for over half an hour then I made a fresh hole and attempted to tempt an ember out. I had wiped off the excess water first and the wood was still damp, however the friction dried out the wood and I was able to tempt out an ember. It took a little bit longer than with dry wood, and the drill did slip a bit at first. The top of the spindle did suffer a bit and bits broke off due to the moisture but the bottom end worked well. Of course, out in the woods, it'll more difficult especially if raining as you'll need to try and be sheltered and keep your hands and your kit dry etc and you'll want to try and place the hearth on something dryish and try not to drip all over it!
The wood also needs to be suitable. e.g right deadness, not green. And some woods ay not work in the damp. A wood like ivy may bot work as that soaks up moisture but woods like dead standing hazel ashould hopefully work in the wilds.