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Target of the bootstrap variances
I have this bootstrap:
library(ggplot2)

n <- 30
set.seed(1)
orig_mean <- 1
orig_sd <- 2
X <- rnorm(n, mean = orig_mean, sd = orig_sd)
set.seed(NULL)

# Bootstrapping
m_reps <- 20000
boot_means <- c()
boot_vars <- c()
for (i in 1:m_reps) {
  current_sample <- sample(X, n, replace = T)
  boot_means <- append(boot_means, mean(current_sample))
  boot_vars <- append(boot_vars, var(current_sample))
}
boot_means <- as.data.frame(boot_means) #for plotting
boot_vars <- as.data.frame(boot_vars)

The central limit theorem tells us that those bootstrapped means $\bar{x}$ should be $\bar{X} \sim N\left(\mu, \frac{\sigma^2}{n} \right)$. We can see this with the plot of $\bar{X} \sim N\left(1, \frac{2^2}{30} \right)$ on top of the bootstrapped means' histogram:
ggplot(boot_means, aes(x = boot_means)) +
  geom_histogram(
    aes(y = after_stat(density)), bins = 20, color = "black", fill = "white") +
  stat_function(fun = dnorm, args = list(
    mean = orig_mean, sd = orig_sd/sqrt(n)))

The bell curve does not precisely line up with the histogram, but it is close.
I would like to do something like this with the variance. The variance histogram is:
ggplot(boot_vars, aes(x = boot_vars)) +
  geom_histogram(
    bins = 20, color = "black", fill = "white")

This is where I'm stuck. Is that a chi-squared distribution? And how do I set up a "target" curve like I did with the bootstrapped means?
What I'm really looking for is to figure out what the distribution is of bootstrapped variances. From there the coding should be fairly straightforward.

See the last sentence of the answer to the linked question for the answer for the type of distribution. Non-statistical coding-type questions are off-topic here, but you should be able to adapt the code you wrote for the distribution of sample mean values to the chi-square distribution associated with sample variances.

This answer might also be helpful.
