Let $\Phi(x)$ be a Gaussian cdf.
Define a function \begin{align} f(x)=\frac{\Phi(x-c)-\Phi(x)}{c} \end{align} for some fixed $c \in \mathbb{R}$.
It is not difficult to show that $f(x)$ is probability density function.
My question does this pdf have a name?
You should have $\frac{\Phi(x+c)−\Phi(x)}{c}$ or $\frac{\Phi(x)−\Phi(x−c)}{c}$; as it stands your quantity has the wrong sign. That, however, is nothing but the average of the Gaussian PDF on $[x,x+c]$ or $[x-c,x]$. So your PDF is the local average of the Gaussian PDF. In fact you could call this a convolution of the Gaussian PDF with a uniform PDF on a certain interval of length $c$. Thus you are looking at the PDF of the sum of a Gaussian and an independent uniform variable on the aforementioned interval.