Can you use a random variable as intput to the Dirac delta, i.e. $\delta(X)$?

Standard random variables have their sample space be numbers, but random variables are much more general than that. You can have random functions, random sets, and in this case, random functionals. A functional is given as an inner product with a function, and outputs a number. In this case, the random functional can have an inner product with a function, and the output is a random number.

The problem here is that $\delta(X)$ is not a function of x, which a delta function usually is. So you would have here $$\langle \delta(X), f(x) \rangle=\int_\mathbb{R} \delta(X) f(x) dx = ???$$ if $X=0$. It would be $= 0$ otherwise.

What is a far more interesting object is $\delta(x-X)$. The math here would yield
$$\langle \delta(x-X), f(x) \rangle=\int_\mathbb{R} \delta(x-X) f(x) dx = f(X)$$ so evaluating $f$ at a random value. There is a rich amount of math that can be done with this object.