The day after the phone call I was able to bring another fan for carpet drying and assess the window situation. A crack between the bricks and the mortar seemed to be the issue. It would also explain why there had to be downpours and pooling water on the sill for it to come inside the house, since the leaks didn't happen every time it rained. Luckily they have plaster walls which are mold resistant, so that's one less thing to worry about, and overall the damage wasn't bad. It still needed to be fixed because a trash bag barrier isn't the classiest.
I happened to be at a Lowes two days later so I picked up some Quikrete Mortar Repair. I already had a trowel from my deck building project and a caulk gun. The weather seemed to be cooperative the next two days with no predicted rain, so the next morning I set to repairing that crack. First I needed to clean it and make sure it was dry. My fan set-up may have been overkill, but I wanted to be sure it was dry after all of the rain we had!
Next up, applying the mortar and smoothing it out. The mortar repair canister actually has a little lip on the end to help get the new mortar into the crack and smooth, but I was glad I had the trowel too. It really helped to make it more even. I probably spent twice as long as I needed to on it, but I feel good about the results. So far there haven't been any more downpours to test it (which means it should be good and "cured" at this point), but crossing fingers that it solves the problem! The before-and-after picture below makes me hopeful that it will!