Why is Water Pooling in My Storm Window Track?

I've noticed that during this rainy weather, water is pooling in the track of my storm window and then seeping down into the window sill. I suspect that the water is collecting in the screen and isn't able to drip outside as it should, possibly because the old windows lack weep holes for drainage. Am I right about the issue, or could there be something else at play? Is there anything I can do myself to resolve this? Thanks in advance for any tips!