Hmmm... good one. DD4 wore FuzziBunz until she trained herself at 20 months. However, we never went on vacation with cloth. You didn't mention what kind of cloth you're using, but I assume you're planning to use an afternoon break for laundry every day? In that case, I think it's possible, but only if you're really willing to carry around multiple poopy diapers all morning. Forgive me if the following is a bit gross, but there's nothing like plain language to get the point across
If the baby is still 100% breastfed, removing poo from the diaper is somewhat optional (as in, if there's a lot, you may want to remove some using a wipe or TP). We never rinsed our diapers until dd was eating solids. Remove diaper, remove insert, pop in washer, leave all day, wash every 24 hours.
If the baby creates more than that, or is eating formula and/or real food, you WILL have to remove the bulk of it. The "cleanest" way to do this is to use a wipe - they are fairly sturdy and won't tear or leave fuzzy residue as TP may. If you use flushable wipes, you can then deposit wipe+poo right in a toilet. Then put the used diaper in a ziploc or other sealable bag. Many people would recommend liberal use of Lysol wipes after "playing with poo". Personally, my cloth diaper lifestyle never really jived with the "everything needs a Lysol wipe" lifestyle. I like soap and water, myself.
I recognize that all these steps are rather inconvenient. The changing table is rarely right next to a sink or toilet. Plus, you'll have a wiggly kid to handle too. If and when available, a family restroom may be best - plus no comments from "well-meaning" people who can't understand the whole cloth thing.
When convenient, it may be easiest to use the Baby Centers at each park. The facilities may be much less crowded. And don't forget to try a few "dry (wet?) runs" at home - take the baby shopping when you think a poo is on the way. Then use the public restroom to find any pitfalls in your plans.
Alternatively, if the baby is fairly consistent, you could try using a disposable ONLY when a bm is expected, and use cloth the rest of the day.
Good luck!