First of all I'd turn it around: the gate being the WiFi host (so active all the time); the car the WiFi client.
Car client switches on when you're ready to drive - i.e. when you start your engine - connects to the WiFi host using stored credentials and opens a URL on the host telling it to open the gate (now, or after a certain delay if it takes time for you to get to that gate).
If your garage or where-ever you park your car is out of range of the gate WiFi it gets tricky. I have no idea on how often the ESP8266 (the MCU of the WeMOS) will try to reconnect if its designated network is not available, and how quick it reacts when the network becomes available (that may be the cause of you having to walk that close to the gate before it reacts - try sitting for a while at distance). That also adds a lot of uncertainty: the distance the WiFi is available may depend on the weather and other rather random parameters.
So basically to make this work you need a reliable way of telling your gate of your imminent approach, at a distance big enough for it to open completely and for you to drive through. You may also want to add a sensor that sees you passing through to close the gate, and a sensor that makes sure no obstacles (you, the car, or the neighbour's cat) are in the gate's way when it closes - and maybe even yet another sensor that will stop the gate if it hits anything while closing or opening.
WiFi may not be the best solution here. Look at 433 MHz links, those are designed for one on one connections and may be more suited for your project.