I am taking the following as the definition of a $\Delta$-complex.
(i) one starts with an indexing set $I_n$ for each $n \in \mathbb{Z}_{\ge 0}$.
(ii) for each $\alpha \in I_n$, one takes a copy $\sigma_\alpha^n$ of the standard $n$-simplex.
(iii) one forms the disjoint union of all of these simplices, for all $n\geq 0$.
(iv) now require that for each $(n-1)$-dimensional face of $\sigma_\alpha^n$, there is an associated $\sigma_\beta^{n-1}$ for some $\beta \in I_{n-1}$.
(v) now form the quotient space by identifying each $(n-1)$-dimensional face of each $\sigma_\alpha^n$ with $\sigma_\beta^{n-1}$ using the canonical homeomorphism. In particular, these homeomorphisms preserve the ordering of vertices.
The sources I've consulted don't make it clear what a subcomplex is. Can someone give me a rigorous definition?
With that (slightly strange way of phrasing the) definition, a subcomplex should be a sequence $(J_n)_{n\geq0}$ such that
for each $n\geq0$ we have $J_n\subseteq I_n$, and
if $\alpha\in J_n$ and $\beta\in I_{n-1}$ are such that $\sigma_b^{n-1}$ is associated to one of the faces of $\sigma_\alpha^n$, then $\beta\in J_{n-1}$.
N.B.: you did not tell us where you got that definition from, nor what other sources you consulted. There are lots of nice expositions of this subject... With time, I've learned to appreciate C. R. F. Maunder's Algebraic Topology, for example.