We explore the constraints on the three-nucleon force (3NF) of chiral effective field theory ($\chi$EFT) that are provided by bound-state observables in the A=3 and A=4 sectors. Our statistically rigorous analysis incorporates experimental error, computational method uncertainty, and the uncertainty due to truncation of the $\chi$EFT expansion at next-to-next-to-leading order. A consistent solution for the 3H binding energy, the 4He binding energy and radius, and the 3H $\beta$-decay rate can only be obtained if $\chi$EFT truncation errors are included in the analysis. All of these except the $\beta$-decay rate give essentially degenerate constraints on the 3NF low-energy constants, so it is crucial for estimating these parameters. We use eigenvector continuation for fast and accurate emulation of No-Core Shell Model calculations of the considered few-nucleon observables. This facilitates sampling of the posterior probability distribution, allowing us to also determine the distributions of the hyperparameters that quantify the truncation error. We find a $\chi$EFT expansion parameter of $Q=0.33 \pm 0.06$ for these observables.