As the classical $(p,q)$-Poincar\'e inequality is known to fail for $0 < p <
1$, we introduce the notion of weighted multilinear Poincar\'e inequality as a
natural alternative when $m$-fold products and $1/m < p$ are considered. We
prove such weighted multilinear Poincar\'e inequalities in the subelliptic
context associated to vector fields of H\"ormader type. We do so by
establishing multilinear representation formulas and weighted estimates for
multilinear potential operators in spaces of homogeneous type.