The 20-year-old, who arrived at Stamford Bridge last week from Chivas Guadalajara, will spend his first year in Europe developing in the Eredivisie.
Vitesse technical director Ted van Leeuwen, who travelled to London to complete the deal, is pleased to have captured the attacking midfielder on loan.
“It took about two months to complete this transfer,” he told Vitesse TV.
“He's very motivated and will fit in perfectly at Vitesse.
“Arnhem is the perfect place to kick-start his European career.”
Prior to joining Chelsea, Davila played for Mexican side Chivas – the same club that nurtured Javier Hernandez through the ranks before he joined Manchester United.
Davila received a call-up to the national squad this summer but has not made his first appearance for the side yet.
He has, however, received 11 caps at Under-20 level, scoring three goals in the process and helping the Mexico team finish third in the Under-20 World Cup this year.