$\DeclareMathOperator\EO{EO}\DeclareMathOperator\SO{SO}\DeclareMathOperator\St{St}\DeclareMathOperator\Sp{Sp}$This is sort of a follow up question to my post here regarding the commutator subgroup of $\SO(5,5,\mathbb{Z})$. As discussed there, this group can be identified with the group of $\mathbb{Z}$ points of a Chevalley–Demazure group scheme of type $D_5$. Now, upon reading a little further into *"Generators, relations, and coverings of Chevalley groups over commutative rings"* by Stein, it seems that we may be able to determine the second integral homology of $\SO(5,5,\mathbb{Z})$ as well. In particular, from Corollary 5.4, we know that if
$$K_2(D_5,\mathbb{Z}) = \ker(\St(D_5,\mathbb{Z})\rightarrow \EO(5,5,\mathbb{Z}))$$
is central, then the second homology of $EO(5,5,\mathbb{Z})$ is isomorphic to $K_2(D_5,\mathbb{Z})$. Here we have let $\St(D_5,\mathbb{Z})$ denote the Steinberg group and $E(D_5,\mathbb{Z})$ the elementary subgroup of our Chevalley group $\SO(5,5,\mathbb{Z})$. As was shown in the linked post, $\EO(5,5,\mathbb{Z}) = \SO(5,5,\mathbb{Z})$.

If our Chevalley group were to be simply connected, then $K_2(D_5,\mathbb{Z})$ is central and has been computed. This is discussed in the introduction to *"The Schur Multipliers of $\Sp_6(\mathbb{Z})$, $\text{Spin}_8(\mathbb{Z})$, $\text{Spin}_7(\mathbb{Z})$, and $F_4(\mathbb{Z})$"* by Stein. From Corollary 3.4 and the remarks at the end of section 4 in *"Stability theorems for $K_1$, $K_2$ and related functors modeled on Chevalley groups"* by Stein, it seems like $K_2(D_5,\mathbb{Z})$ is central. The problem then becomes computing $K_2(D_5,\mathbb{Z})$ in our case.

I'm not sure how to do this. Corollary 4.3.5 of *"Clifford Algebras and Spinor Norms Over a Commutative Ring"* by Bass seems relevant here, but I don't think it applies exactly here.