A plugin (psatz) that ships with Coq in package coq depends on csdp. I have confirmed that it works when package coin-or-csdp is installed. It may make sense to add coin-or-csdp as an optional dependency of coq. Or perhaps as a real dependency -- csdp+blas+lapack is around 3% of the size of coq.

Example file that can be processed iff csdp is installed:

Require Import ZArith Psatz.
Lemma quadratic_Z a c b x (H : (a * x ^ 2 + b * x + c = 0)%Z) : (b ^ 2 >= 4 * a * c)%Z. psatz Z. Qed.

Note that coq caches queries to csdp in .csdp.cache and uses information from that file even if csdp is not installed.