Do this trip a few times a year and have experience with various airlines eg American Airlines, Continental, Aer Lingus. AL used to fly direct to Orlando a few years ago, so we had to use different airlines up to this year. To me the big advantage of AL is the direct flight, particularly in the winter where there can be delays at Chicago, Newark etc, due to weather conditions.
As you intend travelling in the summer, and if you don't mind taking 2 flights instead of one, I would recommend checking fares at
www.kayak.com to see what the best deal is for your party. The system will check all the available flights on your specified dates. However, with 2 children you will probably spend money on food and general shopping while waiting for connection flights. You can also get direct flights to the other Orlando airport, Sanford (about 50 mins from Orlando) if you travel with American Holidays (office in Duke Street) on a package. They don't do flight only deals -
www.americanholidays.com
To finalise, my recommendations are 1) Aer Lingus, 2) Continental or American Holidays charter, 3) American Airlines.