Well to answer the original question, of course it is possible, it's just whether or not you want to.
I'm happy to rewear clothes for a day or two and might be able to manage the clothes aspect, but I do not want to be doing washing or sitting round in laundrettes for half a day on holiday.
I have all the toiletries and suncream that I need at home, that have generally been bought in supermarkets/discount stores. Boots and most shops overseas, seeing as I also don't want to go trailing round looking for reasonably priced versions, are generally extortionate in comparison.
I have loads of 100 ml bottles to dispense shampoo, shower gel and conditioner into, but I'd use 2 or 3 bottles of suncream in a fortnight, just for me, so not possible to take that amount. A family would expect to use 10? bottles? Even with the price difference between supermarket own brand and Boots own brand, that's £30 there.
For the family in the OP, that's why I thought that the best solution would be to take one hold case, which is likely to cost £50/70, not £100+, so they can take all the toiletries they need and have a bit extra space too.
Which if there are men and teens in the party, might be needed, given the size of their shoes, and the fact that most men and teen boys use a lot more clothes than a lot of women as they are more sweaty so cannot rewear things.