In reply to Si dH:
Yes, that would give you a high-quality scenic tour; I can recommend it highly. Most of the huts are now modernised with smaller rooms and showers (Vallonbrun & Leisse may still be fairly modest Park huts, but haven't been to either recently). From recent experience the Plan Sec is particularly friendly and pleasant, with good food. The Arpont has just been enlarged, and is now a bit of a mountain hotel, slightly uber-organised, rather than the rambling old traditional refuge of fond memory.
A lot depends on when you're going - in June it may be worth carrying an axe and a half-rope for the odd remnant hard snow patch here and there, especially if your wife's a bit anxious. From mid-July to mid-August it gets distinctly busy, especially in your first 3 huts which are on the now very popular Tour des Glaciers de la Vanoise. Hut gardien(ne)s are increasingly keen on you booking ahead at any time, to help them with catering; most huts now have websites for advance booking, which in many cases will also help ensure that you can get a 2 or 4 bunk room. But if you speak nicely to them, many hut gardiens will call ahead for you. Most gardiens have a modicum of English, but having even the basic courtesies in French will pay dividends.
I don't think any of the huts have laundry facilities - hot water's at a premium - so not much scope for washing clothes apart from scrubbing things in wash-hand basins or outside water troughs! There are self-service laundrettes in Val d'Isere and Pralognan. Haven't stayed in Val recently, but there is a good range of accomm from chalet-refuge and B&B to decent hotels in Pralo.