Hi Erica; Welcome to Permies!
I suspect that your landlord will not mind you cleaning up the back
yard!
They might even like it a lot!
Some constructive shovel/hoe work might divert some of that standing water.
What about a small garden? Put some of that extra water to use?
Willow and birch
trees both love water, have your little birdy drop them out back!
You can't go wrong improving your rental as long as you do not put up structures or bring in heavy equipment without permission !
Depending how often your landlord stops by they might start suggesting things you could do if you liked...
Good renters are hard to come by. A smart landowner treats a good renter like family.