Should you bring bug spray... hell yes.
Spray (wipe) yourself before you leave the park. The mosquitoes are ALL on the ferry boats, and in the parking lot, with a few at the monorail. They wait for the giant mass of stank sweaty people to provide them with food and water, and a few jump you in your car.
DEET. The higher the % of DEET the longer it lasts, it's like SPF. Again you'll be fine if you spray yourself before you leave the park, but you have to spray all your exposed skin (read directions).
Can you get sick? Maybe, but not likely anything serious. I did lose 24 hours to a Mickey Mosquito bite recently so why chance it. Did I recover? Yes. Was I instructing DW to get ready to call 911, yes. Did she have to? No, thank God, but I would familiarize yourself with the symptoms of encephalitis, you don't want it on your vacation. If you feel like you have the worst flu ever, are confused, hot and cold, with a migraine, nausea, the squirts, and your fever scares you... you might want to see a doctor. Or you could be like me and say, "Honey I think my brain is swelling, if my fever hits 104 and I start convulsing, call an ambulance."
Here's another tip: On the ferry, find a giant spider, frog, lizard, snake, or all 4 and stand near them, the mosquitos will not be near them, except for a few stupid ones.
All in all I've been bitten here about as much as up north so as long as you don't wander into a swamp, lake, Reedy Creek, or jungle after 6pm you're okay. I won't even discuss psorophora ciliata whom I had to punch, slam a door on, and finally take the dirt devil to... then I had to google the thing because I was impressed.