What you would like to do is roughly
C = 1.00;
n = 38;
alpha = 0.05;
Cp = C+0.33;
syms y czero real
fun = chi2cdf((n-1)*(3*Cp*sqrt(n)-y).^2/(9*n*czero^2),n-1).*(normpdf(y+3*(Cp-C)*sqrt(n))+normpdf(y-3*(Cp-C)*sqrt(n)));
solve( int(fun, y, 0,3*Cp*sqrt(n))-alpha, czero)
Unfortunately, chi2cdf does not expect symbolic values, and naively does a test on whether the input < 0 (because input values below 0 give an output of 0), but that test does not work on symbolic values. You can avoid the test by coding chi2pdf in terms of gampdf() but that only postpones the problem, as gampdf() has the same difficulty.
So, instead you have to work numerically:
C = 1.00;
n = 38;
alpha = 0.05;
Cp = C+0.33;
fun = @(y, czero) chi2cdf((n-1)*(3*Cp*sqrt(n)-y).^2/(9*n*czero^2),n-1).*(normpdf(y+3*(Cp-C)*sqrt(n))+normpdf(y-3*(Cp-C)*sqrt(n)));
czero_guess = 0;
czero_sol = fzero( @(czero) integral( @(y) fun(y, czero), 0, 3*Cp*sqrt(n) ) - alpha, czero_guess )
The output I am getting is -1.26021065783808
