A locally compact group whose left and right uniform structures coincide is usually called a SIN group (SIN= small invariant neighborhoods) — see the paper of Itzkowitz I linked to below. It is true that a SIN group is unimodular, see e.g. Theorem 12.1.9 on page 1273 of T.W. Palmer, Banach algebras and the general theory of $\ast$-algebras, Vol. 2, Cambridge University Press, 2001. The reason is that one can find an invariant compact neighborhood $K$ of the identity of $G$ and invariance means that $gKg^{-1} = K$ for all $g \in G$. Therefore
$\lambda(K) = \lambda(gK) = \lambda(gKg^{-1}g) = \lambda(Kg) = \Delta(g) \lambda(K)$
and since $0 \lt \lambda(K) \lt \infty$ we conclude that $\Delta \equiv 1$ and thus $G$ is unimodular.
The other inclusion is false: $\operatorname{SL}_{n}(\mathbb{R})$ is unimodular but its left and right uniform structures are distinct for $n \geq 2$, see Hewitt-Ross, Abstract Harmonic Analysis, I, Example 4.24 (a), p.28f.
Added: It is a result due to G. Itzkowitz, Uniform Structure in Topological Groups, Proc. Amer. Math. Soc. 57, vol. 2, (1976), pp. 363–366 that a locally compact group has a fundamental system of invariant neighborhoods of the identity if and only if the left and right uniform structures coincide.