Have you tried searching for Gore-tex shoes? Clarks makes them for kids, although maybe not of all types nor in all sizes. Maybe Ecco or Geox – although they are more expensive (I think Geox has its own version of gore-tex). You could also try outdoor sports stores like Blacks, Cotswold, Ellis Brigham, or the sites of brands like Columbia, Salomon, etc, but they tend not to be cheap. Maybe Decathlon, although not sure if they’d have gear with proper waterproof membranes.
How wet is the grass? How about leather shoes with no waterproof membrane? Leather has a minimum level of water resistance which may or may not be adequate for you.
Waterproofing a garment is not particularly complex nor expensive. Waterproofing it without causing you to sweat so much as to turn said garment into a cultivation of mould and mushrooms is another story, though.
The cheapest waterproof garments (e.g. most wellies, or kitchen gloves) make you sweat a lot because they block all the airflow and don’t allow any sweat to escape.
The best waterproof garments have membranes (note: membranes, not fabrics) like gore-tex and others, that allow (some) sweat to escape, because they have pores which are small enough to block raindrops but big enough to allow (some) vapour to escape.
Of course we are all different, we all sweat differently, and some people may well be OK with the cheaper waterproofs.
The point is, since waterproof membranes are (relatively) expensive, they’re much rarer on children’s shoes. A good Gore-tex motorcycle or mountain jacket may easily last more than 5 years; a child’s pair of shoes may last a few months before the child outgrows it.