I suggest you to have a look at Zonal Statistics as Table tool. As Hornbydd commented, you cannot explicitly overlay raster and vector data. They should be converted to one or another, either manually or on-the-fly as done by Zonal Statistics as Table.
Make sure you have a shape-area field calculated for your regions. Run the tool by setting zones as your regions shapefile and land cover as the value raster, get the table and join this table to your regions shapefile. And then use AREA field of the table and shape-area from your shapefile to find the proportions in a new field, say COVER_PERC. Finally use Pivot Table on the joined table by setting your region IDs as Input Field, raster values (corresponding to the land cover) as Pivot Table and newly calculated field (COVER_PERC) as Value Field.