The incident dropped both to the back of the field after pitstops.
Germany's Vettel was a point clear of Bottas's team mate Lewis Hamilton going into the first French race in a decade.
Hamilton started cleanly from pole position and led as the safety car was deployed for four laps.
Other collisions on a chaotic first lap at Le Castellet saw two of the three French drivers retire, with Force India's Esteban Ocon and Toro Rosso's Pierre Gasly the unlucky casualties.
Stewards announced an immediate investigation into the collision between Bottas and Vettel, with video replays showing the German had locked up and hit the rear of the Finn's car as they jostled for position.
Bottas returned to the pits for a new front wing while Bottas limped back on three wheels and a rim with a flailing tyre.
"The first corner was nothing to do with me, right?" said Hamilton over the team radio.
Vettel had moved up to 11th by the end of the ninth lap, before the penalty, with Bottas 14th.
(Reporting by Alan Baldwin, editing by Pritha Sarkar)
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