Elsewhere in the MCU, the nearest we have to supporting evidence is Tony's keen sense of intuition about SHIELD agents in Age of Ultron:
CLINT: Company, sorry I didn't call ahead.
TONY: This is an agent of some kind.
(This is the same Tony who thought Natasha Romanov was just a personal assistant who happened to be really good at Krav Maga back in Iron Man 2, but maybe that was the palladium poisoning.)
If we zoom over to the comics, Clint was married to Bobbi Morse, who was also known as both Mockingbird, and Agent 19 of SHIELD.
As the watch we see in Hawkeye also has the number 19 on the back, this seems like a clear implication that in the MCU Laura Barton was an agent of SHIELD. (Maybe the 19th? It's not clear how the numbering works.)
Previously in the show, Clint told Kate that the watch belonged to "a friend" he used to work with, who'd been out of the game for a long time, and that the watch could compromise their identity and put them at risk. Although Laura is more than his friend, this fits fine with what little we already know about her, and her remark to Clint about how careful "they'd" been.
(In Agents of S.H.I.E.L.D., which may or may not be canon for the MCU, a character named Bobbi Morse was played by Adrianne Palicki, but was (I believe) never referred to as Agent 19, and had no connection to Hawkeye.)