Yes. Convert the grasslands data to raster and, if necessary, query that to produce a binary indicator raster of grasslands presence (1 or logical 'true' where grasslands exist and 0 or logical 'false' where they do not). The focal (neighborhood) mean of this binary raster produces a "simple" density map. Use a circular neighborhood of the desired radius.
More sophisticated densities, based on other kernels, can be computed using weighted focal means. If this proves too onerous (ArcGIS does not offer strong capabilities for such calculations), any software that performs convolution--including most image processing software--can handle this calculation very quickly even on large rasters.