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I agree with @misolee: not to get the park strollers, the ones we saw were plastic and hard, not ideal for napping.

We went when DS was 11 mos old and technically still on two naps. He fell asleep in the stroller or in the carrier and we just went with it. He even fell asleep during some of the shows (that were very very loud) but sleep comes easier bc they are so tired.

We adhered to our normal bedtime as much as possible and let them sleep in /wake naturally to avoid over tiredness. It worked our fine for us and I was very worried about a sleep deprived, cranky baby. But he did great. No real issues or melt downs at all.

Another vote to skip the park strollers. The sun shade isn't great and like others said they are hard plastic seats that don't recline.
We took our daughter at almost 3 and she napped both days in the park in her stroller for nearly 2 hours. When it was around her normal time, we just laid her back with her usual sleepy things (blanky, lovey, and put the shade way down. Then we kind of lazily strolled around some quieter areas of Magic Kingdom and did minimal check-ins so she would get bored/be able to fall asleep. Then we strolled for a bit to look at things, and ended up taking a nice break at a quiet bench with a snack for ourselves.
The next day at Epcot we did the same thing and used her nap time to get all the snacks at the World Showcase during the Food and Wine festival and sat at the biergarten area in Germany and have a drink.
I see you're going with family too, so it might also be an option for somene else to look after the napping little while you two get to ride a rollercoaster or two if that's your thing. We were with my in-laws and at MK they told us to go off and have some fun while she napped so we also took some time to hit up Space Mountain (our fave) and meet back up with them.
We left the park both days around 5 or 6 to enjoy dinner elsewhere and get a semi-reasonable bedtime. I didn't think she would enjoy the fireworks at that age anyway and it would have been way too long of a day for all of us.

I miss when she would nap at the park honestly, it was a nice break for everyoneon! Later on when we've taken her after naps stopped, we like to go back to the hotel mid-day to escape the heat and use the pool or chill at the room in the AC for a bit. Then we go back around/after dinner to catch a few more rides/attractions and stay for fireworks or night time shows.

Have fun!!

The Disney World strollers suck and are $$$. I always rent from Kingdom Strollers for the week and have had great service.

In my experience, kids crash wherever, whenever at Disney World because the days are so busy. Around nap time, I recline the seat, close the sunshade and take a little walk around the park....it never takes long for them to fall asleep!

@schubr03: yes definitely, our kids will be 3 and 5 next time I go and we will still use a double stroller

@schubr03: Rent a double stroller. You will not regret it. There's no way your 3 year old will be able to keep up all day with all the walking.

@schubr03: Agreed with the double stroller option (to bring or to rent). We brought a stroller for my LO just last year when she was nearly 5 because it's just so much distance/walking. It's also very, very hot so it would be even harder to keep up with walking, especially at the speed you sometimes need to take to get from point A to point B for fast passes, show times, or food reservations.

@schubr03: def do a double.. we went w our 4 yo and 11 mos old and they both rode in it a LOT... majority of the time.