How do I align peak labels?

hjust = 0.5 should work better. Using hjust = 0 aligns your labels a little to the right, with the top edge of the text aligned with the middle of each peak.

Here's a reproducible example that doesn't rely on external data that might not remain available at that link. (See bottom for application to the OP dataset.)

library(ggpmisc)
library(ggrepel)
library(ggplot2)
x <- data.frame(wv = 2300:2000)
x$abs = abs(cos(x$wv/50) * sin(x$wv/2))
ggplot(x, aes(wv, abs)) +
  geom_line() +
  stat_peaks(colour = "black", span = 11,  
             geom ="text_repel", direction = "y", 
             angle = 90, ignore_threshold = 0.09, 
             size = 3, x.label.fmt = "%.2f", 
             vjust = 1, hjust = 0.5, segment.color = "red") +
  scale_x_reverse(limits = c(2300,2000))

Here's loading the original data:

library(readr)
x <- read_csv("~/Downloads/CO-FTIR Spectrum-1800 mTorr-2021.csv")
colnames(x)=c("wv", "abs")

Here with these parameters added:

box.padding = 0.0, nudge_y = 0.02,

I wasn't able to find a way to do this without constructing an auxiliary dataset. However, once constructed the actual plotting then becomes fairly easy:

The code is as follows:

library(dplyr)
library(ggplot2)
library(ggpmisc)
library(ggrepel)
library(pracma)


#####################################
# Constants
#####################################
wvfilter = c(1975, 2275)
data_filename = "data.csv"


#####################################
# Read data
#####################################
dat = read.csv(data_filename)
colnames(dat)=c("wv", "abs")


#####################################
# Identify peaks
#####################################

# Extract peaks
peaks = data.frame(findpeaks(dat$abs, threshold=0, minpeakheight=0.03))
# Give data frame reasonable names
colnames(peaks) = c("height", "x_index", "peak_begin_index", "peak_end_index")
# Convert from index to wavelength
peaks$wv = dat$wv[peaks$x_index]
# Set the y position of the labels
peaks$nudge = 0.3-peaks$height
# Filter by wavelength
peaks = peaks %>% filter(wvfilter[1] <= wv & wv <= wvfilter[2])


#####################################
# Plot data
#####################################

ggplot(x, aes(x=wv, y=abs)) +
  geom_line() +
  geom_text_repel(
    data=peaks,
    mapping=aes(x=wv, y=height, label=height),
    force=0,
    nudge_y=peaks$nudge,
    direction="x",
    angle=90,
    segment.color="red"
  ) +
  xlab(bquote('Wavenumbers ('~cm^-1*')')) +
  ylab("Absorbance (A.U.)") +
  scale_x_reverse(limits=rev(wvfilter), expand=c(0,0)) +
  ylim(-0.01,0.29) +
  ggtitle("FTIR - Carbon Monoxide, Fundamentals")

And this gives the plot:

Some notes:

• In the image I'm using the wrong text for the labels. Fix this with label=wv instead of label=height.
• The peaks dataset needs to be filtered using the same values passed to scale_x_reverse(limits=. To simplify this, I've introduced a global variable for setting the x-axis limits.
• Playing with the force argument gives different degrees of clustering of the labels. force=0 seems to work best for your data.
• You need to manually set the peaks$nudge value. I don't know of a good way of doing this other than guess-and-check.

Here's an approach using ggplot2::stage and the built-in segment and text geometries:

library(ggplot2)
library(ggpmisc)

id <- "1S345TaPqANriDPLN6H_PyunuM1QHi485"
x <- read.csv(sprintf("https://docs.google.com/uc?id=%s&export=download", id))
colnames(x)=c("wv", "abs")

ggplot(x, aes(x=wv, y=abs)) + geom_line() +
  xlab(bquote('Wavenumbers ('~cm^-1*')')) + ylab("Absorbance (A.U.)") + ggtitle("FTIR - Carbon Monoxide, Fundamentals") + ylim(-0.01,0.29) +
  stat_peaks(mapping = aes(x = stage(wv, after_scale = x + 0.25)),
             geom ="text", colour = "black", span = 11, angle = 90,
             ignore_threshold = 0.09, size = 2.5, x.label.fmt = "%.2f",
             vjust = 0.5, hjust = -1) +
  stat_peaks(mapping = aes(xend = stage(wv, after_scale = x + 0.25),
                           y = stage(abs, after_stat = y + 0.005),
                           yend = after_stat(y + 0.02)),
             geom ="segment", lwd = 0.5, colour = "red", 
             span = 11, ignore_threshold = 0.09) +
  scale_x_reverse(expand=c(0,0), limits = c(2275,1975))

From help(stat_peaks) we can see that x and y stats are calculated for each peak. Typically we can access these stats with after_stat, but because you've also transformed the x-axis, you need to actually access that stat after it's been scaled.

In the first layer of stat_peaks we use stage(wv, after_scale = x + 0.25) to add a bit of nudge to the x coordinate. In the second call of stat_peaks we set the geometry to segment. Segment requires an xend and yend. We can use stage and after_stat to again add a nudge and line length.