In my honest opinion....maybe.
Personally, I wouldn't want to go to WDW in June -- too crowded for my taste. (Yes, I know many, many people go then and have a lovely time.) Also, I wouldn't want to take my two children on my own. But my kids are a very active 3.5 year-old and an equally active 11 month old.
Consider the activity and temperment of your twins. Are they going to be tearing you in two directions at once? Or are they pretty easy to handle?
I would get a few of the planning books from the local library. You can read about the rides and get a good idea which rides the kids would like. Many rides at WDW can handle 3 people at once. If your children are slender, you might be able to squeeze into a ride that is meant to fit two adults like Buzz Lightyear (intended to fit two adults, but we've fit 2 adults and a toddler). There are also a lot of shows to which you could take the girls.
Personally, if I were you, I would get the wrist-strap harnesses for busy times (like exiting Fantasmic, etc.). It's so easy to get separated from one child during a high vollume event, little own two children. It will be much more comforting to have one child strapped to each wrist. (The wrist straps connect one of your wrists to one wrist of your child. I've never had one, but heard about them here.)
Also, at that time of year, I would make Priority Seating reservations for all of my meals in the park. The last thing you want is to be standing in a long counter service line with two children and no one to help you hold the food, keep track of the kids, etc... It would be much easier to sit down for meals and have someone bring your food to you.
Also, you'll probably still want to take the twins back to the hotel for a nap in the afternoon. Staying at a WDW resort makes taking an afternoon break much easier.
I hope this helps!
EthansMom