The 21-year-old, whose final appearance for the non-league club came in the FA Cup first-round defeat to Northampton on Sunday, has agreed a two-and-a-half year deal at London Road.
Posh moved for the striker after he netted 16 goals in as many games for the Skrill South outfit.
Former United striker Dwight Gayle had a loan spell with Bishop's Stortford ahead of his move to Peterborough.
Vassell is available for Posh immediately, due to the transfer window not covering non-league football.
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