user32906: All of the wiring diagrams I've found for wiring up a GY-521 accelerometer to an Arduino have looked something like this:

I've read in several places that it needs to be connected to A4 and A5. I want to connect one to a SparkFun ThingDev, which unfortunately only has a single analog input:

Is this possible? 
I only need it for a very simple application (finding out if a machine is vibrating or not).

user4143: A4 and A5 on the Arduino double as the I2C pins SDA and SCL.  You are communicating with I2C, not analog.
The ESP8266 doesn't have I2C in hardware, but there is a software-based I2C implementation which can work on any pins.  The pinout you show has I2C labeled on pins 2 and 14. While you can use those you could use any two pins of your choosing.
Note that I2C on the ESP8266 has a slightly different begin function:
Wire.begin(2, 14);

That is where the pins you want to use are assigned - in this case, pin 2 for SDA and pin 14 for SCL.

32906: <p>All of the wiring diagrams I've found for wiring up a GY-521 accelerometer to an Arduino have looked something like this:</p>

<p><a href="https://i.stack.imgur.com/mA0LF.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/mA0LF.png" alt="enter image description here"></a></p>

<p>I've read in several places that it needs to be connected to <code>A4</code> and <code>A5</code>. I want to connect one to a SparkFun ThingDev, which unfortunately only has a single analog input:</p>

<p><a href="https://i.stack.imgur.com/x1G2A.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/x1G2A.png" alt="enter image description here"></a></p>

<p>Is this possible? </p>

<p>I only need it for a very simple application (finding out if a machine is vibrating or not).</p>
 4143: <p>A4 and A5 on the Arduino double as the I2C pins SDA and SCL.  You are communicating with I2C, not analog.</p>

<p>The ESP8266 doesn't have I2C in hardware, but there is a software-based I2C implementation which can work on any pins.  The pinout you show has I2C labeled on pins 2 and 14. While you <em>can</em> use those you could use any two pins of your choosing.</p>

<p>Note that I2C on the ESP8266 has a slightly different begin function:</p>

<pre><code>Wire.begin(2, 14);
</code></pre>

<p>That is where the pins you want to use are assigned - in this case, pin 2 for SDA and pin 14 for SCL.</p>
