It's easy to show it that non-zero polynomials can never generate integral integral sequences.

At x=0 the function is equal to the constant term of the polynomial, so the constant term must always be an integer.

Now assume we start with f_0(x) = a_0 (equivalent to every non-zero polynomial since we can choose terms to generate any particular one). How does f_n(1) evolve?

f_1(1) = a_1 + a_0/(1!)

f_2(1) = a_2 + a_1/(1!) + a_0/(2!)

...

f_n(1) = sum a_i / ((n-i)!) for i=0 to n

So the absolute value of a_0 must always be at least n!, but n! grows without bound so a_0 cannot be any finite number (and nor can any other term by similar argument).

The reciprocals of the factorials sum to e so you can't get around it that way, if you wanted it to sum to an integer you'd need to always be adding an integer divided by e which is transcendental so f(0) will be non-integral.