We consider the point-to-point log-gamma polymer of length $2N$ in a half-space with i.i.d. $\operatorname{Gamma}^{-1}(2θ)$ distributed bulk weights and i.i.d. $\operatorname{Gamma}^{-1}(α+θ)$ distributed boundary weights for $θ>0$ and $α>-θ$. We establish the KPZ exponents ($1/3$ fluctuation and $2/3$ transversal) for this model when $α=N^{-1/3}μ$ for $μ\in \mathbb{R}$ fixed (critical regime) and when $α>0$ is fixed (supercritical regime). In particular, in these two regimes, we show that after appropriate centering, the free energy process with spatial coordinate scaled by $N^{2/3}$ and fluctuations scaled by $N^{1/3}$ is tight. These regimes correspond to a polymer measure which is not pinned at the boundary. This is the first instance of establishing the $2/3$ transversal exponent for a positive temperature half-space model, and the first instance of the $1/3$ fluctuation exponent besides precisely at the boundary where recent work of arXiv:2204.08420 applies and also gives the exact one-point fluctuation distribution (our methods do not access exact fluctuation distributions). Our proof relies on two inputs -- the relationship between the half-space log-gamma polymer and half-space Whittaker process (facilitated by the geometric RSK correspondence as initiated in arXiv:1110.3489, arXiv:1210.5126), and an identity in arXiv:2108.08737 which relates the point-to-line half-space partition function to the full-space partition function for the log-gamma polymer. The primary technical contribution of our work is to construct the half-space log-gamma Gibbsian line ensemble and develop, in the spirit of work initiated in arXiv:1108.2291, a toolbox for extracting tightness and absolute continuity results from minimal information about the top curve of such half-space line ensembles. This is the first study of half-space line ensembles.