A lot depends on the amount of rain, what the rain is doing, time of day, and cover type around me.
I will not hunt a heavy rain or with heavy rain close to hitting. I define heavy as something that will wash out a light but reasonable blood trail in less than 3 hours. I will not take a shot with very much rain at all late in the day. I will hunt a light rain or clearing weather in morning particularly when the leaves thin out where I have a real good chance of getting on a trail before it is gone, can see the animal moving longer distances and can see well to spot other tracking clues.
I had to hunt a lot of rainy weather and wind this year just to get out much. It was one of those years for me on my Friday and Saturday hunting schedule. I did pass on shots four times this year due to heavy rain setting in. I chose to just set in a blind and watch / scout some days.
Last year I shot a deer with light rain starting about 1-hr before dark. The doe jumped the string and I got a forward angling hit behind the shoulder as she dropped and turned away. The blood trail was too light to follow in the rain and dark. Next day blood was long gone. I had to grid search the next day. Never found her and totally disrupted a great beading area sending a great buck for my area off the property for the rest of the year from what I could tell. That was all around a bad deal. I had shot one the year before under almost exactly the same conditions from the same stand with better results. That one ran in a better direction where it was a lot easier to find, but still had to grid search for that one due to no blood trail. In the past I just didn't hunt in the rail, but bad weather timing the last few years has me hunting a little more in less than ideal conditions. I got a lot more conservative again this year. It cost me shots, but then I didn't loose anything either.