A question about reading and calculating thermal resistance of a voltage regulator from its datasheet
You have calculated the power dissipated in the wind gauge - what you need to know is the power dissipated in the LM317.  The wind vane will draw 1 mA, and you have 3 volts across the LM317, for a power dissipation of 3 mW due to the load. However, the LM317 requires a minimum load current of 5 mA for proper operation - the voltage-setting resistors are usually selected to draw that current.

With 1 mA for the wind gauge, and 5 mA for the LM317 operation, the LM317 will dissipate about 6 mW - no need for any heat sink at such a low power.  You could use a TO-92 packaged LM317 without concern.