Hetty58 makes a good point, it should have been her partner, the father of her children who moved out of their family home, not her and the children. I do realise separation with two such young children isn't something a young mother would do without need.
Is the house on mortgage? Can you help your daughter pay her share of the mortgage, with a legal agreement to ensure her ex partner doesn't make merry with any financial gains that accrue. If rented, he should move out and let her and their children have continued stability in what has been their home. That way, you can offer practical and maybe financial support and she keeps her children and her dogs.
Losing the dogs will compound her feelings of loss of control, loss of what she thought was a life long, loving, shared parenting relationship. Devastating for her.
If none of that is possible, I honestly feel you will struggle to find what will be long term foster carers for these dogs. Most breed specific and any other dog charity will be familiar with placing dogs who come into their care because of relationship breakdown. They are unlikely to be in a position to offer fostering for an unlimited, unspecified timescale. Dogs Trust, Breed specific charities, local rescue shelters can all offer to rehome her dogs. I can't imagine losing my dogs, as any stage of my life, especially when I was a young mother with a child (yes, I was that person). However, the children have to come first. that's just how it is. Either she regains what was the family home and lives there with children and dogs, with what sounds like excellent support from you, or she gives the dogs up and moves in with you on what has to be a temporary basis.
best of luck with this difficult situation