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With this syntax you are trying to create a callable symbolic expression. It doesn't work because the symbolic sum that you are using requires a symbolic expression (depending on the symbolic summation index m here) as the first argument, and wigner_3j in Sage does not accept symbolic arguments, only numeric arguments (more precisely, only integers or half-integers). This explains the errors. (By contrast, Sage does support symbolic binomial coefficients and sums, and hence callable symbolic expressions involving them.)

In this case you have no need for symbolics at all; you can just define a plain Python function, using the plain (non-symbolic) sum:

wigner_4j = lambda j1,j2,j3,j4,j,m1,m2,m3,m4: sum(wigner_3j(j1,j2,j,m1,m2,m)*wigner_3j(j,j3,j4,-m,m3,m4) for m in range(-j,j+1))

Or, without lambda:

def wigner_4j(j1,j2,j3,j4,j,m1,m2,m3,m4):
    return sum(wigner_3j(j1,j2,j,m1,m2,m)*wigner_3j(j,j3,j4,-m,m3,m4) for m in range(-j,j+1))