USB host mini boards with MAX3421E chip are pretty cheap. Search Ebay, aliexpress, etc. for "usb host mini board". I do not know of any bit bang implementation of USB host.
The USB Host Shield library supports ESP8266 and ESP32.
Another option is the SAMD21 supports USB host mode so a cheap SAMD21 board such as an Adafruit Trinket M0 might work. Connect the Trinket M0 to the ESP8266 via UART to transfer the VID/PID. A USB OTG to host adaptor or cable is required.
Here is an example project of a Trinket M0 running in USB host mode.