Puglia or Sicily with toddler

I really don't think any factor matters except which one you would rather visit--you won't see all of either one. Time of year is not a big factor, so I'd let flights dictate the choice.
If you go with Sicily, you can still make smaller moves, you don't have to traverse the island. OTOH, If you would rather save Sicily for when you can do more sightseeing, that is a valid reason (but with two littles, that might be a few years). The sights in Puglia are less intensive--a small village here, a church or piece of art there, not go-go-go like Sicily can be (but does not have to be).
If you go with Puglia, I'd stay a week in the Salento and a week near/in the Itria valley.

I was thinking of basing in two places in Puglia as you mentioned. Where would you stay in Sicily? We’d like to visit Ortigia, Taormina, Cefalu and Palermo as well as beaches.

We were also trying to pick a destination that is stroller friendly with not too many steps and heard that some placed in Sicily are hilly, like Taormina.

Also, going to Puglia is a bit easier as we can go to Naples with a direct flight then rent a car and drive ~3 hours. With Sicily, we have to do a layover to get there as there are no direct flight.

Thanks for your help!

I have not been to Sicily yet but I plan trips often--whether I take them or not, lol.
For Sicily, I would plan around the major cities and choose either east or west--so plan around Palemo (to include Cefalu) or Catania (to include Taormina and Ortigia). I would try to resist the urge to do both sides but with two weeks you certainly could. It's just my impulse to concentrate on a smaller area.

Are you aware of sandy beaches on the east side? I heard Cefalu has nice beaches which is why I wanted to explore that area as well! I will check plane tickets to fly into one city and fly out of another if it makes sense. Would you stay in Ortigia or near Taormina ?

Others will come along with good Sicily info.
If you go to Puglia, sandy beaches are more commonly on the Ionian side, with notable exceptions like Torre Guaceto and Torre Rinalda on the Adriatic.

I have been to Sicily but not Puglia. Sicily is harder logistically to get to. That for me would favor Puglia.

I do though have a friend who went to Sicily with two young children for two weeks. They went to Ortigia, Toarmina, and somewhere near Ragusa’s beach.

The beaches at Cefalu are not fabulous. Those near Ragusa are much better.

I have been to both regions.

The driving in Puglia is some of the easiest I have undertaken in Italy. While driving in Sicily is not nearly as stressful as many reports suggest, Puglia is easy easy easy. Get a good Puglia guidebook and get excited about the destination. National Geographic has a good Puglia guide - National Geographic Traveler: Puglia.

Lots of tourists in either place. (We loved Sicily but it is a very big island.)

https://www.nationalgeographic.com/travel/article/puglia-salento-slow-pace-makes-it-one-of-italys-best-road-trips

@BethFL the direct flight to Naples then us driving to Puglia feels less overwhelming than a layover and a second flight to Sicily.. so it’s a big reason we’re leaning towards Puglia. In Sicily it would be impossible to base in one strategic place?

@Elena my husband will drive and he’s an expert so thankfully we’re not worried about that. He’s driven all over Europe. But it’s a bonus that it’s easy! Where did you stay in Puglia? There will be driving every day for sure

We were thinking of driving back to Naples on the day of our flight but the flight would be at 11:30 AM which means we would need to be up and driving by 4-5 AM… it sounds a bit intense. Any recommendations?

If you don't want to spend the last night in Naples, where driving would be ill-advised, you could consider staying in much more manageable Salerno. What would you consider the max amount that you would prefer to drive on departure morning?
Not sure I'd go through all of that for a direct flight though--did you see what the schedules looked like for Bari and Brindisi? (If you see something that looks tolerable, I think that would be easier, ultimately. Munich is not too bad for layovers, but I would avoid Frankfurt if possible.)