We get a lot of rain here. I keep the food in the coop, to keep it dry. As long as the chickens have a dry coop to go into, they should be fine.
A tarp over at least part of the run can keep a heavy deluge off the chickens, so they can still be in the run. If the mud isn't too bad, you could add some sand to the run. If it's really saturated or there are lots of puddles all the time, you may want to get a wood pallet and put it in there, to give the chickens something dry to get up on.
For a run that is having problems with normal weather, I would work on modifying it. If this is only a very rare occurrence, maybe you don't want to do a lot of work on improving drainage or regrading. There are lots of threads on muddy runs, if you search on that. They have a lot of good ideas for solutions.