Around 400 filling stations - among the 11,000 filling stations in France - had no fuel at all on Wednesday morning. And another 800 had dwindling stocks, according to trade unions and the consumer website mon-essence.fr.
The authorities said no fuel depots had been blocked but several were operating at reduced capacity.
The strike was being felt at a growing number of filling stations, exacerbated by panicking drivers rushing to fill up their vehicles ahead of a holiday weekend.
The Ufip oil industry federation urged drivers not to besiege filling stations.
Government spokesman Christophe Castaner echoed the appeal, saying the pumps "have all been resupplied as normal" but that panic-buying was causing problems.
A year ago, workers striking against labour market reforms blocked access to fuel depots across France, forcing the government to dip into oil reserves to ease shortages in the build-up to the Euro 2016 football championships taking place in the country.
In the current strike, tanker drivers want their working day to be limited to 10 hours and are also demanding increased medical checks and pay due to the often hazardous nature of the products they are transporting.
"Our aim is to have a meaningful sign from the authorities before deciding how the strike unfolds," said Fabrice Michaud of the transport branch of the CGT union.
"We also hope to convince other unions to come to the negotiating table," he added.