I'm a neonatal/peds nurse. Babies can have seizures if given too much water because it throws off their electrolytes. Giving them sips between bites of solid food is ok, but I wouldn't give them more than a few ounces a day.
It's impossible to give an age that's ok to start water, because it really depends on their diet more than their age. If it's extremely hot and you are worried about dehydration, you can give some pedialyte, or even half pedialyte and half water mixed, but limit the plain water.
Remember, even if the baby is hot and sweaty, they are not going to dehydrate as fast as you because they are sitting in a stroller and you are walking possibly up to several miles a day while on vacation. The main thing is keeping them as cool as possible. Keep them shaded from the sun, and take a washcloth that you can dampen with cold water to wipe them down. Only use a sun hat when they're in direct sun and need it to keep them from burning. If you leave it on all the time, it will not allow their body heat to escape from their head and they will get overheated quickly. Get a fan for your stroller, or a sqirt bottle fan. I'm sure older siblings would love spritzing the baby with water and faning them! If your baby is too young for sunscreen, don't forget thin socks. I always worry about babies in strollers who's bodies are shaded by the canopy, but their little bare feet are sticking out into the sun.