Your company should normally charge Italian VAT on your sales unless the turnover relative to French customers exceeds the threshold of € 100.000. In that case, your company must be VAT registered in France and charge French VAT.
Voluntary VAT registration in France can be also envisaged by your company even if the threshold is not exceeded. This option is interesting in order to apply the French VAT rate [20%] lower than the Italian VAT rate [22%] on your sales.
Companies selling goods via internet to private individuals [B2C] must pay attention to the following issues: