First question, are you driving or if flying renting a car? I would never suggest an off site stay of any kind without a car! If you don't have a car I would suggest that you stay on site at both resorts (WDW & Universal- I would skip
Legoland without a car) and uber or take a cab to move from WDW to Universal and then from Universal back to the airport.
I'm also a family of five. We have three teens 13,16,18 and are WDW annual pass-holders for several years. We live about 2 hours from WDW and are vets at both Universal and WDW. We have done many on and off site visit to both WDW and Universal. We have done Legoland once (day trip) it was cool but once was enough for us. Yet, keep in mind my children are slightly older then yours.
If you have a car I would consider staying off site in a condo. Space, two bathrooms (or more), a washer/dryer, and full kitchen are so nice when you have three children! We still eat out a lot when we travel this way, it's still a vacation, but having food available for my kids (2 are boys) makes life so much more pleasant for all of us. Space also = sanity for me! Are you looking more for a resort type stay with theme pools and activities or just a clean, safe, place to sleep and shower?
If you want a clean, safe basic condo with a nice pool area I would suggest Windsor Hills. We are booking a 3 bedroom, two bath for a week in July (high season) through Vacation Central Florida for $854 total with all fees and taxes. We have stayed here many times an I highly recommend them. Vacation Central Florida best deals on condo rentals are typically Sat- Sat rentals but not always again ask. We're staying Sun-Sun in July. The location would work well for you as its in the middle of all three parks. It's approximately 10-20 minutes to WDW, 25-30 minutes to Universal and about 40 minutes to Legoland. There are several condo rentals places all around the Orlando area. Just make sure you check out fees things like cleaning fees and minimum stay requirement fees. I always ask about all fees and factor these into the total price when comparing prices.
Another option might be to stay off site for most of the stay and then either before or after the condo stay check in onsite at Universal. I would do my last night at Universal, basically packing to go home the night before and have one or two carry on bags only to take into the Universal resort hotel and leave the rest in the car. It sounds like a hassle but its worth a big perk as
most onsite Universal resorts offer front of the line passes for hotel guest. They are good the day you check in until the day you check out. Basically, this means you can check in the hotel early am (room is not ready yet), get your front of the line passes do one day at each Universal park (or hop back and forth if you buy hopper passes) by only staying one night here. Also, there are a few rides in the Harry Potter areas that don't offer Front of the line passes but one park opens early each day for onsite hotel guest. We hit these rides first thing and then do the rest of the parks with very little wait. Those front of line passes are wonderful in busy times like summer, Christmas and Spring breaks. Do know that Universal, unlike Disney, does charge for parking at their hotels even if staying there and you will be charged for parking per day so one nights stay is charged parking for two days. Also, you may have 5 guest in their standard rooms but they charge an extra $25 a night for a roll away bed and will require it for 5 people in one room.
Just be prepared going in that this is a different type of vacation then just doing WDW and staying on site. Neither, in my opinion, is better then the other they're just different. Condo stays most times don't come with daily maid service and requiring driving. Yet, the ability for space and privacy at what is usually lesser cost even with parking and car rental (if priced right) balances out. It's also nice to see what else Orlando has to offer besides Disney.
Good Luck, in my opinion, there is no wrong way to do a Orlando vacation you just have to go what works best for your family.