Early welcome to OKC! I hope you have a good move, that is a long trip! I don't know of any rental companies here but I do know a few great real estate agents if you need one of those in the future. One of our local news stations paired up with our newspaper and the have a lot of classified ads and listings here: https://newsok.com/classifieds. Just click on the left side of the page on rentals and it should drop down and show you a ton of options like apartments, houses, and duplexes with the area of OKC they are in.
Hopefully this will help some and get you started! Congrats on the new little girl!