Neither one of my kids will nap in the stroller. Of course, now my DS who is 3.5 won't nap at all. We find that it totally reenergizes us to head back to the hotel for an afternoon break.
On past trips with just our son, I would take him back to the room and lay down for a nap with him while DH stayed at the parks and did the adult rides, then he'd meet us back at the hotel in the late afternoon. We would have dinner and then head back to the parks for a parade or show.
On our upcoming trip -- first one with our 1 year-old daughter, we'll all head back to the hotel in the afternoon and I'll get DD down for a nap while Dad and DS get some pool time in.
Our basic schedule is that we're at a park when it opens or closely thereafter, do attractions, have lunch in the park, do a few more attractions and then head back to the hotel about 2ish, nap till 4-4:30, get some dinner (either at our resort or at a park), a few more attractions and a night show (like Fantasmic). We usually seem to miss the parades which are in the afternoon, but between missing a parade and a nap, I'd choose to miss the parade.
If you're staying off-property, it may be easier to find a quiet place for your children to rest at the parks. At MK, my son would nap in a heartbeat on the monorail and you can ride it in circles as long as you want -- just take something to read.
Best of Luck
EthansMom (and now Molly's too)