You can certainly attach $L$-functions to half-integer weight eigenforms, but you don't get anything really new by doing so: they turn out be versions of $L$-functions of integer weight modular forms. More specifically, there is the "Shimura lifting" map from weight $k + 1/2$ to weight $2k$, which sends eigenforms to eigenforms; and the L-function of a half-integer weight eigenform will be closely related to that of its image under the Shimura lift. See e.g. here:
www.math.wisc.edu/~ono/reu06shimura.pdf
In fact this turns out to be a very powerful way of studying the L-functions of integer weight forms (used, for instance, in Tunnell's work on the congruent number problem, which uses modular forms of weight 3/2 to understand the values at $s=1$ of the $L$-functions of twists of elliptic curves.