A flooded backyard can be handled with a sump pump or a PVC drainpipe with a Beaver Underground Downspout.
You can send most of that trapped water to the front yard without the sump pump. A PVC drainpipe from the back yard to the front could carry the surface water along with water from the gutters. The pipe might not be buried that deep considering the pitch is only 1-inch per ten feet.
Before a job of this size is considered, some landscape engineering is required. How low is the backyard compared to the front? This question is answered with a line or water level.
First, set some 36-inch stakes, one in the front yard, one on the side of the house, and another in the back yard. Attach the line or water level to a mark on the middle stake. Mark the stake in the front and back with a mark that is level to the center stake. Stretch a string from stake to stake at the level marks to set up a level reference line.
Measuring from the string to the ground will indicate if the front yard is sufficiently lower than the back to provide adequate drainage without the use of a pump. The collector of water in the back yard will be a trench with about ten-feet of drain tile (PVC pipe with holes in it) surrounded with pea stone.
Fill the trench with stone up to the surface so water can easily drain to the pipe. It will not be a summer before the grass overcomes the surface stone.
Once the collector is built, and it can be level and as large as needed, use solid pipe the rest of the way to the front yard.
If all goes well, and the pipe can achieve a pitch of one-inch per ten-feet or more you can attach the gutters that are near the pipe when you go past the house. This will take some rain runoff out of the back yard.
Out in the front yard at the end of the drainpipe, install a Beaver Underground Downspout. The top pops up when water discharges out of it during heavy rains. When installing it use at least a bag of pea-stone around and below the head. This will aid dispersion of small amounts of water from light rains.