one way Fisher's Exact Test I can use a chi square test for a one way table like this:
frequencies <- c(12, 3, 9, 19, 3, 14, 13)
chisq <- chisq.test(frequencies)
chisq 

However, when I do something similar for the Fisher's Exact Test like this:
fisher.test(frequencies)

It results in:
Error in stats::fisher.test(x, y, ...): 'x' must have at least 2 rows and columns
Traceback:

I am a bit confused. Can one not do a one way Fisher's Exact Test? Thanks.
 A: For fixed total number of observations
If the total number of observations is fixed, then the sample distribution of the observations, given the null hypothesis, will be a multinomial distribution and an exact test can be the "multinomial test".
You can compute this exactly, approximate it by using a Monte Carlo approach, Pearson's chi-squared test, or with a likelihood ratio (this latter one relates to a G-test).
In R the package XNomial can do all of them
library(XNomial)

obs <- c(12, 3, 9, 19, 3, 14, 13)
n <- sum(obs)
k <- length(obs)
p <- 1/k
predicted <- rep(n*p,k)
xm <- xmulti(obs,predicted)

Giving
$$\begin{array}{lcl}
p_{\chi^2} &=& 0.002994365 \\
p_{\text{Likelihood ratio}} &=& 0.001218835 \\
p_{\text{exact}} &=& 0.001233232
\end{array}$$
This package uses some corrections for the statistics or the p-values. If you would compute the values manually then you get the same statistics but slightly different p-values.
obs <- c(12, 3, 9, 19, 3, 14, 13)
n <- sum(obs)
k <- length(obs)
p <- 1/k
predicted <- rep(n*p,k)

numerator <- factorial(n) * p^n
logdenominator <- sum(lfactorial(obs))
p_obs <- numerator/exp(logdenominator)

### MonteCarlo
set.seed(1)
nmc = 10^6
counts = 0
for (i in 1:nmc) {
  x <- rmultinom(1, n, rep(p,k))
  sample <- sum(lfactorial(x))
  if (sample >= logdenominator) {counts = counts+1}
}
counts/nmc

### Pearson's Chi-squared
chi2 <- sum((obs-predicted)^2/predicted)
1-pchisq(chi2,k-1)

### Log Likelihood ratio
f_obs <- obs/n
LLR <- sum(obs*log(predicted/obs))
test_ratio <- -2*log(p_theory/p_obs)
1-pchisq(ratio,k-1)

