Yes, this is true, but the result is still being written up by Geelen, Gerards and Whittle as part of their Matroid Minors Project.

Also, for any fixed signed graph $H$, there is a polynomial-time algorithm to test if an input signed graph contains $H$ as a minor. Together with the positive answer to your question, this implies that there is a polynomial-time algorithm to test for any minor-closed property of signed graphs. See my PhD thesis for the algorithm (which works more generally for any $\Gamma$-labelled graph, where $\Gamma$ is a fixed finite abelian group).

In the thesis there is also a mention of the well-quasi-ordering result that you want (Theorem 1.1.9).