Yes, you can use Bookvault alongside either of these POD providers.
Things to consider;
If you have used a free ISBN from KDP, you could only use that to place orders directly on the Bookvault portal, on our store The Great British Book Shop and selling on your own website.
There will be no conflict of having books distribute by IngramSpark & Bookvault, the main bookshops and retailers such as Gardners will send the order to whoever is listed as the distributor with the ISBN registry (Nielsen in the UK). Other marketplaces that we could sell on such as Amazon or Alibris, we will show as another sales option alongside IngramSpark.