The man, who was thought to be armed with a gun, entered the branch in Bridlington Street shortly before 9.40am this morning.
After threatening staff, he escaped with around £350 in cash.
He was seen getting into the driver's side of a grey Nissan car, possibly a Micra or Qashqai, and then drove off in the direction of Bridlington.
North Yorkshire Police said a full investigation had begun into the robbery, which left staff shaken but unharmed.
A spokeswoman said: "The search for the man is ongoing. Any witnesses, or anyone with information which may help the investigation is asked to contact police."
Officers are particularly keen to speak to any residents or business owners in the Bridlington Street area who may have captured the incident on CCTV.
Call police on 101, select option 1 and speak to the Force Control Room, quoting reference 12170123935.
Alternatively, Crimestoppers can be contacted anonymously on 0800 555 111.