Average response time of Esp8266 module
I think response time would be a big problem. I used the ESP8266 with a temperature and humidity sensor, complete with a small web server on the ESP8266, and it works fine. But I wouldn't want to control an RC car with it. But if you buy it on ebay from China, it is only a few dollars, so buy one anyway.

Use Bluetooth, either a couple of HC-05 modules, or the Bluetooth Low Energy HM-10 modules. Both can communicate at 115200 baud. The HM-10 modules may be a bit tricky, because some Chinese sellers put a stripped down version of firmware on the module, and forces you to search online how to flash a usable firmware version.