Deer can eat grass, but it's not their preferred food, and they can't thrive on it. They eat forbs, shoots, browse (twigs, buds, etc.) and mast like acorns (they are set up to deal with the large amounts of tannin in acorns).

https://www.msudeer.msstate.edu/deer-diet.php

"Although low quality forages such as mature grasses provide adequate nutrition to animals such as elk and cattle, the quicker digestive process of whitetails requires more readily digestible forages to fulfill their energy and protein requirements. On severely overpopulated and depleted ranges, white-tailed deer have starved to death with their stomachs full of low quality forages."