Once you derive this term:
$$
k(\vec x, \vec y) = e^{-\gamma ||\vec x - \vec y ||^2}
$$

you get terms that only depend on $\vec x$, terms that only depend on $\vec y$, and terms that are mixed.

The terms that separate nicely, aren't a problem, as you could write them as a dot product. 

e.g. in the simple case where x is 1-D:

$$
k(\vec x, \vec y) = e^{-\gamma ||x - y ||^2} = e^{-\gamma (x^2 -2xy + y^2)} \\ 
e^{-\gamma (x^2 + y^2)} = e^{-\gamma (x^2)} e^{-\gamma (y^2)}
$$

Here you see the separate terms can be separated into a (dot) product.

The terms that are mixed are a problem - but the trick is to use Taylor series to separate them into an infinite series of products:

$$
e^{-2\gamma xy} = \sum_{k = 0}^{\infty} \frac{(2\gamma xy)^k}{k!} 
$$

Here $x, y$ are products. 
So the gaussian kernel can be written as the dot product between the following vectors:

$$
<e^{-\gamma (x^2)}, 1, \sqrt{2 \gamma}x, \frac{2 \gamma x^2}{\sqrt 2}, ...> \cdot <e^{-\gamma (y^2)}, 1, \sqrt{2 \gamma}y, \frac{2 \gamma y^2}{\sqrt 2}, ...>
$$