Most resorts have them at check-in, but I’m not sure for the Swolphin - maybe try the guest services desk there (inside the gift shop, I think depending on the time you’ll find the guest services cast members either on the Dolphin side or the Swan side).
Alternatively, you can get the buttons at guest services at the entrance of any park
This!
For me personally,
this would have been heavenly when my kids were still young enough to make you feel like you are literally on duty 24/7. I’m one of those rare people who doesn’t enjoy spa time or even massage - but if your wife is into that, the Dolphin has one of the few spas on property and it gets good reviews.
Today (kids are 9 and 11) I think I would ask for a fancy character breakfast at Topolino’s if I happened to be in the World with my family for Mother’s Day.
Same 
Do you think they can stroller nap? My kids passed out easily in the stroller as they got exhausted. That’s helpful when they do.. I know they wouldn’t have napped if we went back to the hotel as they rarely napped in beds at home ( had to go for a drive) plus the excitement.
Also, I didn’t realize the extra challenge of finding stroller parking - that can take time as you can only park in designated areas and it will get moved if you are illegally parked. Put a ribbon on your stroller to help ID it and maybe an AirTag.
Absolutely true for my kids. “Nap now” was close to impossible particularly in hotels.
The nanny idea is great advice and wasn’t even on my radar. How does that work? Are there reputable Disney-managed services?
Never have done it, but someone else was talking about it recently.