Hecke eigenforms whose eigenvalues at $p$ are $p$-adic units are called p-ordinary. The dimension of the space spanned by $p$-ordinary eigenforms over many reductive groups, including Siegel modular forms, is known to be bounded regardless of levels and weights. Furthermore, Hida showed that the dimension is bounded in the case of half-integral weight modular forms as well, using Shimura correspondence. We will show the similar result for half integral weight Siegel modular forms of degree 2, using an analogy of Shimura correspondence for Siegel modular forms (Ibukiyama conjecture).