How can I work out gram staining information from the species name?
The GOLD database can be used to retrieve a list of species names according to their Gram staining: go to the "Advanced search" page and select the either "gram-" or "gram+" under "Organism Fields" -> "Gram Stain". Then on the results page click on the number under "Organisms". On the results page you can select to have 1000 results per page, meaning that you would need to do some manual work.

I should add that as far as I know this information is based on what people submit when they register their genome projects, so I wouldn't expect this list to be curated or 100% accurate.

If you have access to the genome of the species you are interested in (you could download a representative strain for each species using ncbi-genome-download), you could use traitar, which can predict the gram staining, among other interesting broad phenotypes.