In the context of principal components analysis (PCA), the bootstrap is commonly applied to solve a variety of inference problems, such as constructing confidence intervals for the eigenvalues of the population covariance matrix $Σ$. However, when the data are high-dimensional, there are relatively few theoretical guarantees that quantify the performance of the bootstrap. Our aim in this paper is to analyze how well the bootstrap can approximate the joint distribution of the leading eigenvalues of the sample covariance matrix $\hatΣ$, and we establish non-asymptotic rates of approximation with respect to the multivariate Kolmogorov metric. Under certain assumptions, we show that the bootstrap can achieve the dimension-free rate of ${\tt{r}}(Σ)/\sqrt n$ up to logarithmic factors, where ${\tt{r}}(Σ)$ is the effective rank of $Σ$, and $n$ is the sample size. From a methodological standpoint, our work also illustrates that applying a transformation to the eigenvalues of $\hatΣ$ before bootstrapping is an important consideration in high-dimensional settings.