You answered your own question. Let $F:\text{Alg. Sp.} \to \text{Sets}$ be the functor that associates to every algebraic space $T$ the set of equivalence classes of triples $(\mathcal{L},\mathcal{V},q,p)$ of an invertible sheaf $\mathcal{L}$ on $T$, a locally free sheaf $\mathcal{V}$ on $X\times T$, a homomorphism of coherent sheaves on $X\times T$, $$ q:\text{pr}_X^*M\otimes_{\mathcal{O}_{X\times T}} \text{pr}_T^*\mathcal{L} \to \mathcal{V},$$ and a homomorphism of coherent sheaves on $X\times T$, $$ p:\mathcal{V}\to \text{pr}_X^* L,$$ such that the following sequence is a short exact sequence, $$ 0 \to \text{pr}_X^*M\otimes_{\mathcal{O}_{X\times T}}\text{pr}_T^*\mathcal{L} \to \mathcal{V} \to \text{pr}_X^*L \to 0. $$ An equivalence from the triple $(\mathcal{L},\mathcal{V},q,p)$ to a second equivalence $(\mathcal{L}',\mathcal{V}',q',p')$ is a pair $(\alpha,\beta)$ of an isomorphism $\alpha:\mathcal{L}\to \mathcal{L}'$ of invertible sheaves on $T$ and an isomorphism $\beta:\mathcal{V}\to \mathcal{V}'$ of locally free sheaves on $X\times T$ such that both $p'\circ \beta$ equals $p$ and such that $\beta\circ q$ equals $q'\circ \text{Id}_M\otimes \alpha$.
The functor of equivalence classes of triples is represented by the universal object $\mathcal{U}$.