I can get the web to be plotted with the following code.
library(bipartite)
data <- read.csv("Dillon_Data.csv")
str(data)
'data.frame': 186 obs. of 3 variables:
$ Site : int 1 1 1 2 2 2 2 2 2 2 ...
$ Plant : Factor w/ 30 levels "Achillea millefolium",..: 30 30 29 19 12 9 9 9 22 22 ...
$ Pollinator: Factor w/ 29 levels "Apis mellifera",..: 4 1 5 25 5 1 4 13 4 4 ...
Site_list<- frame2webs(data, varnames = c("Plant", "Pollinator","Site"),
type.out = "list", emptylist = TRUE)
plotweb(Site_list$`1`)

I guess this is exactly what you are doing. So I suggest, saving your work and restarting R, you can restart R by pressing Control, Shift, F10. This should clear all the objects out of your global environment and detach all the loaded libraries. Then try and run the above code.
If that doesnโt work, I would then try make sure that R is up to date. The installR package will allow you to update R. You can update R Studio from the Help tab and then check for updates.
To make the plots look nicer you can look at all the parameters there are to play with, run ?plotweb().

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