They do have rush. I believe it is either $40 or $49, and most of the seats tend to be side orchestra with a little obstruction, according to the man I spoke to at the box office.
The Producers (OBC), The Wild Party (Lippa Original Cast), Miss Saigon (Lea Salonga), Anything Goes (Patti LuPone), Assassins (Original Cast), Gypsy (Patti LuPone), The Secret Garden (OBC), Pippin (Mostly OBC), Mean Girls (Pre-Broadway), Miss Saigon (Original Replacement Cast), Parade (OBC), The Civil War (OBC), Guy And Dolls (1992), A Doll's House Part 2, The King and I (Faith Prince), The King and I (Donna Murphy), Blood Brothers (O
For the 2015 Color Purple Revival, I was able to see Cynthia Erivo, and both her understudies Bre Jackson and Adrianna Hicks as Celie. I got to see Jennifer Hudson, Heather Headley, Jennifer Holliday and Carla Stewart as Shug Avery. And Danielle Brooks and Carrie Compere as Sofia.