I don’t think you need a culvert unless there is a stream? Can’t see the sides but this looks like a classic low spot high traffic drainage issue. You need to fix the low spot where water is accumulating. I would have someone come out with a bobcat, remove all the dirt and mud and haul it away, and fill it in with gravel/limestone dust. Put down geotextile if you can swing it, but it will be OK if not as long as you get to the hard stuff and scrape out all the mud.Mound it so water drains away. Take the stone about 20 feet out in all directions to help with traffic wear and tear, more if there is ground showing already. Basically anywhere there is wear, cover with gravel.
This cannot be fixed economically by hand, the bobcat at least is absolutely necessary. I think that would be better than something bigger because you can do so much detail work with them and this is a small space. Best of luck.
Ask around, my landscaper/handyman has a bobcat and runs it on my farm for $35 an hour. Cheap! And he is very good at driving it which I am not. You can probably find someone like him pretty easily. Lots of landscapers own them.