Is there a continuous Taylor/MacLaurin transform (like the Fourier transform)?

The Laplace transform does exactly what you're talking about. The polynomials don't necessarily come in the form ${1 \over n!} x^n$ but can be substituted into that form.

Check out this lecture by Arthur Mattuck from MIT for the details.

I've never thought about it as a generalization of the Taylor series, but what you're describing sounds like the Mellin transform, which is defined as

$$\hat{f}(s) = \int_0^{\infty}x^{s-1} f(x)\mathrm{d}x$$

It's used a lot in certain branches of high-energy physics theory (which is how I happen to know about it). The Wikipedia article linked above describes some other uses.