Ooh I worked for eurocamp on that campsite one summer. And did lots of eurocamp holidays as a child.
For bedding, if you don't order bed sheets you just get a blanket per person. What we always did as children was take sleeping bags and pillow cases, and my parents took a duvet etc.
I would take tea towels and a sharp knife for the kitchen, as the knives aren't that great. Also take soap and washing up liquid, toilet paper, kitchen roll, bin bags.
You can hire pushchairs (but they don't lie flat), cots (they are travel cots and won't fit in your room), bed rails, and high chairs.
That site is big, and has a very steep hill on one side. The pool is good, although almost all of it is outside unless it's changed. The shop is small but reasonably well stocked and (again, unless it's changed) there's fresh baked bread/croissants on site each day as well. Le Bugue is not far (10-15 min drive) and is a nice town with a big supermarket so you can get everything you need there anyway. The restaurant La Pha in Le Bugue was lovely, it did amazing duck and the garden backed onto the river - obviously this recommendation is about 10 years old, but it's worth looking up on tripadvisor to see if it's still recommended.
Having worked for eurocamp I would stay there, but I would wash the kitchen stuff (pans, plates etc) before I used it.
If you need extras of anything in the caravan/tent (which are you staying in?) then if you ask, they should hopefully let you have it. Eg if you find with a small child you'd like a couple of extra tumblers so you aren't washing up as often. Depending on the type of accommodation you'll have one tumbler and one wine glass per person it sleeps I think.