I would like to know if it's possible to send data from a sensor to an arduino wirelessly. The arduino and sensor won't be very far apart (a meter at most). I know I would probably need more electronics but do keep in mind that I'd like to keep the sensor side as small as possible.
Yeah that would be possible with RF module XBee as it has analog and digital pins on it with which you can interface sensors and then can use X-CTU software to configure and send the data to 2nd XBee which will be attached with the Arduino UNO.
You will need to design a small power supply circuit as XBee works on 3.3V, you can LM317 for that purpose.
Maybe you could connect the sensor to an ESP8266 and use another ESP8266 at the receiving end? The ESP8266 is a microprocessor with WiFi. And if it just need to communicate with another ESP8266 you can use the simpler ESP-NOW rather than WiFi