Definitely do-able. We did it last year with slightly older children in August and we cant wait to go back.
Here's a brief list of what we enjoyed, but please let me know if you want any more details:
Look at Accor chain of Hotels. We stayed in an Ibis Styles - very family friendly with lovely affordable food. We actually ate there most mealtimes in the end and just bought a makeshift picnic at lunchtime wherever we ended up.
The Metro system was very easy to use around the city once you understand the system (v quick to understand), but I don't know how easy it is with a buggy, we struggled with suitcases on the first and last days.
The Batobus (hop on hop off boat) was brilliant - but hot, take bottles of water if the sun is out, it's a bit like a greenhouse on the water.
Go to the Jardin d'Acclimatition rather Disney. Much better (and cheaper) for little ones. Take a change of clothes as they will get wet in the water fountains. Though our kids had dried off before we got round to changing. Crocs or similar footwear that will dry too. Take a picnic.
Pre-buy timed tickets for the sights you must do. We did this the Eiffel Tower as it was top of the kids list. It was great not to queue.
We steered clear of the shops and aimed for the parks. Some near famous landmarks and some not. I particularly enjoyed the parks in the evening as the whole of Paris comes out to play. If we went again, I would pack some balls and racquets as we never managed to buy any, though we did find a big Decathlon near to the Josephine Baker Swimming pool when we needed to buy swimming caps and budgie smugglers.
Sure I will think of more... it was a great holiday 