Yep you can use your own rubber dinghy there are lots of places. Nearest is probably Borrans Park www.lakedistrictwiki.co.uk/Borrans_Park you can easily walk from Ambleside, have a picnic etc.
You could also drive out to any number of local lakes Rydal, Grasmere most if not all within 45 minutes.
The other place I would recommend is Brockhole, an old house in enormous lakefront grounds, my favourite place as a child with an enormous play area - outdoor - lots of activities especially over the summer holidays. I think they have just opened a family wildlife centre too. It's about a 10 minute drive from Ambleside, free of charge but you do pay for parking. The other option would be to take a boat from Waterhead, nr Ambleside although I'm unsure if your dog would be allowed on the boat. Plenty of space for the dog at Brockhole and can even going swimming in the lake. The cafe also has outdoor seating so again your dog will be fine, if the rain stops by then!
Slightly further a field in Bowness - 4 miles, again you could drive or get the boat, is The World Of Beatrix Potter, I think your children would enjoy this if you want an indoor activity.
Let me know if you want any more ideas