The 28-year-old former Salisbury player joined Rovers in 2014 and has since made 20 first-team appearances.
He played in the promotion final win against Grimsby which earned Rovers promotion back to the Football League.
Puddy joins Braintree, 22nd in non-league's top flight, as competition for Sam Beasant.
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National League side Braintree Town have signed goalkeeper Will Puddy on a one-month loan deal from League One club Bristol Rovers.