I assume you have a layer with the county boundaries. Export this layer as a CSV file, keeping only the name of each county in the export step, and don't export the geometry.
Now open this file in your spreadsheet app of choice.
Add a new column for each plant species.
Put a one in that column for each county where the species is present.
Save the file (still as CSV).
Open the county layer and the new CSV files in Ggis.
Open the properties window for the county layer and click on the Joins tab.
Click on the "+" to add a new join. Join to the CSV layer, and select the county names field as the join field for each.
Now your county layer has the plant species column.
To display a single species, open the properties window for the county layer, select symbology. Set the symbology as rule based. To color just those counties with species 1, your rule would say something like Species1 = 1, where Species1 is the field name you used. Pick a color/transparency of your choice, then set the "all others" rules to be the background color you want.
That does it for one species. You can then export to tiff or whatever. Getting it to cycle through all the different species is for someone else to explain.
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