@Dirty-flow already provided a good answer but note that some ski resorts occasionally extend their opening dates later in the season depending on snowfalls. It’s mostly for locals/week-end skiers as advertising it in advance and having people book expensive trips would be too risky at that time of the year.
Mid-March on the other hand should really be no problem. In fact, it’s often the best time: the days are already a little longer, the bigger crowds from the school holidays are gone (in France this year they take place between February 15 and March 17 depending on the region), some rentals are slightly cheaper than during the high season’s peak and with a bit of luck the snow cover can still be very good. The risk of having too little snow is higher than in February but I have had many great skiing days in that period of the year.
Regarding Morzine specifically, I haven’t been there recently so I don’t remember it very well but judging by the map and altitude, I suppose it might be impossible to go all the way down to the village by ski late in the season. You would then need to go up and down the mountain with the gondola but that’s not a big deal. Avoriaz is higher and should be no problem. Of course, bad weather (either too hot/no snow – even artificial – or clouds/rain making skiing unpleasant or downright impossible) is always possible so it’s never a 100% thing.