Drainage issue, water causing foul odor in home

What is the geology of the area you are in. I am assuming the ground is fairly flat since you imply the water outside is causing the inside to have a smell.

If you have access to the ground / yard, you could dig a rock pit. Dig a pit, put filter fabric to line the pit and fill it with rock, then put the grass back on top. This gives the water an easy place to drain away - depending on how much water and how quickly water drains through soil back into the aquifer it could help.

Do you have carpets? What is your flooring type over the concrete?

I’d guess there is a sewage problem that occurs when it rains. (Rain will cause septic tanks to overflow, or cause waste to spew out a broken pipe, etc. and could flood to the surface.)

Instead of you trying to fix the problem, I’d take a small glass jar, clean and sterilize it, take a sample, give it to a local lab (you can contact the DEQ or local sanitary department for name and address of suitable lab), and then have them test for “fecal matter”.

If it tests positive, I’d give it to your landlord. If he refuses to do anything, I’d give the results to the local DEQ office, health department, etc.

If it tests negative, then I’d contact a local geologist, perhaps at a nearby college, to see if there is something in the soil that smells when it rains.

In a comment, you mentioned that the apartments were built on top of a "swamp area". Swamps smell like, well, swamps.

When it rains it's bringing up the smells from the swamp land below and the wetlands nearby. It's not that the apartment management isn't willing to do anything about it, there isn't anything they can do about it.

Unfortunately, you're going to have to move away from the source of the odors or talk the city/county/environmental groups into paving over the wetlands, and good luck with that.