Since e^(x^2) is a continuous function, yes, it HAS an integral (anti-derivative). Every continuous function (and many non-continuous functions) is the derivative of some function and therefore has an anti-derivative.
Is that anti-derivative any "elementary function" (defined as polynomials, rational functions, exponentials, logarithms, trig functions and combinations of them)? No, if fact for most functions the anti-derivative is not an elementary function. (There are more functions in heaven and earth than are dreamed of in your philosophy, Horatio!)
Of course one can always DEFINE a new function to do the job. I don't know specifically about e^(x^2) but the ERROR FUNCTION, Erf(x) is defined as an anti-derivative of e^(-x^2).